On the signed chromatic number of some classes of graphs
Julien Bensmail, Sandip Das, Soumen Nandi, Théo Pierron, Sagnik Sen, Eric Sopena
aa r X i v : . [ c s . D M ] S e p On the signed chromatic number of some classes of graphs
Julien Bensmail a , Sandip Das b , Soumen Nandi c ,Théo Pierron d,f,g , Sagnik Sen e , Éric Sopena d a Université Côte d’Azur, CNRS, Inria, I3S, France b Indian Statistical Institute, Kolkata, India c Institute of Engineering & Management, Kolkata, India d Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France e Indian Institute of Technology Dharwad, India f Faculty of Informatics, Masaryk University, Botanická 68A, 602 00 Brno, Czech Republic g Université Lyon 1, LIRIS, UMR CNRS 5205, F-69621 Lyon, France
Abstract
A signed graph ( G, σ ) is a graph G along with a function σ : E ( G ) → { + , −} . A closedwalk of a signed graph is positive (resp., negative) if it has an even (resp., odd) numberof negative edges, counting repetitions. A homomorphism of a (simple) signed graph toanother signed graph is a vertex-mapping that preserves adjacencies and signs of closedwalks. The signed chromatic number of a signed graph ( G, σ ) is the minimum number ofvertices | V ( H ) | of a signed graph ( H, π ) to which ( G, σ ) admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention in the lastdecades, especially due to their strong connections to the theories of graph coloring andgraph minors. These homomorphisms have been particularly studied through the scopeof the signed chromatic number. In this work, we provide new results and bounds on thesigned chromatic number of several families of signed graphs (planar graphs, triangle-freeplanar graphs, K n -minor-free graphs, and bounded-degree graphs). Keywords: signed chromatic number; homomorphism of signed graphs; planar graph;triangle-free planar graph; K n -minor-free graph; bounded-degree graph.
1. Introduction
Naserasr, Rollová and Sopena introduced and initiated in [15] the study of homo-morphisms of signed graphs , based on the works of Zaslavsky [20] and Guenin [9].Over the passed few years, their work has generated increasing attention to the topic,see e.g. [2, 6, 8, 14, 16, 19]. One reason behind this interest lies in the fact that homo-morphisms of signed graphs stand as a natural way for generalizing a number of classicalresults and conjectures from graph theory, including, especially, ones related to graph mi-nor theory (such as the Four-Color Theorem and Hadwiger’s Conjecture). More generally,signed graphs are objects that arise in many contexts. Quite recently, for instance, Huangsolved the Sensitivity Conjecture in [10], through the use, in particular, of signed graphs.His result was later improved by Laplante, Naserasr and Sunny in [11]. These interestingworks and results brought yet more attention to the topic.In the recent years, works on homomorphisms of signed graphs have developed followingtwo main branches. The first branch of research deals with attempts to generalize existingresults and to solve standing conjectures (regarding mainly undirected graphs). The sec-ond branch of research aims at understanding the very nature of signed graphs and theirhomomorphisms, thereby developing its own theory. Since our investigations in this paper
Preprint submitted to ... September 28, 2020 re not related to all those concerns, there would be no point giving an exhaustive surveyof the whole field. Instead, we focus on the definitions, notions, and previous investigationsthat connect to our work. Still, we need to cover a lot of material to make our motivationsand investigations understandable. To ease the reading, we have consequently split thissection into smaller subsections with different contents.
Throughout this work, we restrict ourselves to graphs that are simple , i.e., looplessgraphs in which every two vertices are joined by at most one edge. For modified typesof graphs, such as signed graphs, the notion of simplicity is understood with respect totheir underlying graph. Given a graph G , as per usual, V ( G ) and E ( G ) denote the set ofvertices and the set of edges, respectively, of G .A signed graph ( G, σ ) is a graph G along with a function σ : E ( G ) → { + , −} calledits signature . For every edge e ∈ E ( G ) , we call σ ( e ) the sign of e . The edges of ( G, σ ) in σ − (+) are positive , while the edges in σ − ( − ) are negative . In certain circumstances, itwill be more convenient to deal with ( G, σ ) in such a way that its set of negative edges isemphasized, in which case we will write ( G, Σ) instead, where Σ = σ − ( − ) denotes the setof negative edges. Note that the notations ( G, σ ) and ( G, Σ) are equivalent anyway, since σ can be deduced from Σ , and vice versa .Signed graphs come with a particular switching operation that can be performed on setsof vertices. For a vertex v ∈ V ( G ) of a signed graph ( G, σ ) , switching v means changingthe sign of all the edges incident to v . This definition extends to sets of vertices: for a set S ⊆ V ( G ) of vertices of ( G, σ ) , switching S means changing the sign of the edges in thecut ( S, V ( G ) \ S ) . For S ⊆ V ( G ) , we denote by ( G, σ ( S ) ) the signed graph obtained from ( G, σ ) when switching S . Two signed graphs ( G, σ ) and ( G, σ ) are switching-equivalent if ( G, σ ) can be obtained from ( G, σ ) by switching a set of vertices, which we write ( G, σ ) ∼ ( G, σ ) . Note that ∼ is indeed an equivalence relation.An important notion in the study of signed graphs is the sign of its closed walks. Recallthat, in a graph, a walk is a path in which vertices and edges can be repeated. A closedwalk is a walk starting and ending at the same vertex. A closed walk C of a signed graphis positive if it has an even number of negative edges (counting with multiplicity), and negative otherwise. Observe that the sign of closed walks is invariant under the switchingoperation. In fact, the two notions are even more related, as revealed by Zaslavsky’sLemma. Lemma 1.1 (Zaslavsky [20]) . Let ( G, σ ) and ( G, σ ) be two signed graphs having the sameunderlying graph G . Then ( G, σ ) ∼ ( G, σ ) if and only if the sign of every closed walk isthe same in both ( G, σ ) and ( G, σ ) . Before moving on to all the definitions and notions related to signed graph homomor-phisms, let us point out to the reader that the main difference between signed graphs and -edge-colored graphs lies in the switching operation. Recall that a -edge-colored graph ( G, c ) is a graph G along with a function c : E ( G ) → { , } that assigns one of two possiblecolors to the edges, but with no switching operation. Thus, in some sense, -edge-coloredgraphs stand as a static version (sign-wise) of signed graphs. It was noticed in [7, 15, 19]that homomorphisms of -edge-colored graphs are closely related to homomorphisms ofsigned graphs. For the sake of uniformity and convenience, we below refer to such ho-momorphisms as sign-preserving homomorphisms of signed graphs. The study of suchhomomorphisms was initiated in [1] independently from the notion of homomorphisms ofsigned graphs. 2 sign-preserving homomorphism (or sp-homomorphism for short) of a signed graph ( G, σ ) to a signed graph ( H, π ) is a vertex-mapping f : V ( G ) → V ( H ) that preservesadjacencies and signs of edges, i.e., for every uv ∈ E ( G ) we have f ( u ) f ( v ) ∈ E ( H ) and σ ( uv ) = π ( f ( u ) f ( v )) . When such an sp-homomorphism exists, we write ( G, σ ) sp −→ ( H, π ) .The sign-preserving chromatic number χ sp (( G, σ )) of a signed graph ( G, σ ) is the minimumorder | V ( H ) | of a signed graph ( H, π ) such that ( G, σ ) sp −→ ( H, π ) . For a family F of graphs,the sign-preserving chromatic number is generalized as χ sp ( F ) = max { χ sp (( G, σ )) : G ∈ F } . A signed graph ( H, π ) is said to be a sign-preserving bound (or sp-bound for short) of F if ( G, σ ) sp −→ ( H, π ) for all G ∈ F . Furthermore, ( H, π ) is minimal if no proper subgraph of ( H, π ) is an sp-bound of F .We are now ready to define homomorphisms of signed graphs. It is worth mentioningthat the upcoming definition is a restricted simpler version of a more general one [17].A homomorphism of a signed graph ( G, σ ) to a signed graph ( H, π ) is a vertex-mapping f : V ( G ) → V ( H ) that preserves adjacencies and signs of closed walks. We write ( G, σ ) → ( H, π ) whenever ( G, σ ) admits a homomorphism to ( H, π ) .The next proposition highlights the underlying connection between sp-homomorphismsand homomorphisms of signed graphs. This proposition actually provides an alternativedefinition of homomorphisms of signed graph. Proposition 1.2 (Naserasr, Sopena, Zaslavsky [17]) . A mapping f is a homomorphismof ( G, σ ) to ( H, π ) if and only if there exists ( G, σ ′ ) ∼ ( G, σ ) such that f is an sp-homomorphism of ( G, σ ′ ) to ( H, π ) . Just as for sp-homomorphisms, the signed chromatic number χ s (( G, σ )) of a signedgraph ( G, σ ) is the minimum order | V ( H ) | of a signed graph ( H, π ) such that ( G, σ ) → ( H, π ) . For a family F of graphs, the signed chromatic number is given by χ s ( F ) = max { χ s (( G, σ )) : G ∈ F } . Moreover, a bound of F is a signed graph ( H, π ) such that ( G, σ ) → ( H, π ) for all G ∈ F .A bound of F is minimal if none of its proper subgraphs is a bound of F . One can observe that if ( G, σ ) is a signed graph having positive edges (resp., negativeedges) only, then ( G, σ ) → ( K χ ( G ) , π ) , where χ ( G ) denotes the usual chromatic numberof the graph G and ( K χ ( G ) , π ) is the signed complete graph of order χ ( G ) having positiveedges (resp., negative edges) only, and thus χ sp (( G, σ )) = χ s (( G, σ )) = χ ( G ) . Hence,the notions of sign-preserving chromatic number and signed chromatic number are indeedgeneralizations of the usual notion of chromatic number.For undirected graphs, homomorphism bounds of minimum order are nothing but com-plete graphs. The study of sp-bounds and bounds for signed graphs is thus much richerfrom that point of view, as one of the most challenging aspects behind determining χ sp ( F ) or χ s ( F ) for a given family F can actually be narrowed to finding (sp-)bounds of minimumorder.One hint on the general connection between the sign-preserving chromatic number andthe signed chromatic number is provided by the following results.3 emma 1.3 (Naserasr, Rollová, Sopena [15]) . Let ( G, σ ) and ( H, π ) be two signed graphs.If ( G, σ ) → ( H, π ) , then, for every ( H, π ′ ) ∼ ( H, π ) , there exists ( G, σ ′ ) ∼ ( G, σ ) such that ( G, σ ′ ) sp −→ ( H, π ′ ) . The connection between the sign-preserving chromatic number and the signed chro-matic number was shown to be actually even deeper, through the concept of double switch-ing graphs. Given a signed graph ( G, σ ) , the double switching graph ( ˆ G, ˆ σ ) of ( G, σ ) isobtained from ( G, σ ) by adding an anti-twin vertex ˆ v for every vertex v ∈ V ( G ) , whichmeans that for every uv ∈ E ( G ) , the graph ˆ G contains the edges uv, u ˆ v, ˆ uv, ˆ u ˆ v and theirsigns satisfy σ ( uv ) = ˆ σ ( uv ) = ˆ σ (ˆ u ˆ v ) = ˆ σ ( u ˆ v ) = ˆ σ (ˆ uv ) . One connection between ( G, σ ) and ( ˆ G, ˆ σ ) is the following. Theorem 1.4 (Ochem, Pinlou, Sen [19]) . For every two signed graphs ( G, σ ) and ( H, π ) ,we have ( G, σ ) → ( H, π ) if and only if ( G, σ ) sp −→ ( ˆ H, ˆ π ) . In particular, this result implies the following relations between the two chromaticnumbers.
Proposition 1.5 (Naserasr, Rollová, Sopena [15]) . For every signed graph ( G, σ ) , we have χ s (( G, σ )) ≤ χ sp (( G, σ )) ≤ χ s (( G, σ )) . An even deeper connection, based on (sp-)isomorphisms of signed graphs, was estab-lished by Brewster and Graves [7]. A bijective (sp-)homomorphism whose inverse is alsoan (sp-)homomorphism is an (sp-)isomorphism . Two signed graphs are (sp-)isomorphic ifthere exists an (sp-)isomorphism between the two.
Theorem 1.6 (Brewster, Graves [7]) . Two signed graphs ( G, σ ) and ( H, π ) are isomorphicif and only if ( ˆ G, ˆ σ ) and ( ˆ H , ˆ π ) are sp-isomorphic.1.3. Our contribution In this paper, we establish bounds and results related to the sign-preserving chromaticnumber and the signed chromatic number of various families of graphs. More precisely, wefocus on planar graphs with given girth, K n -minor-free graphs, and graphs with boundedmaximum degree. Each of our results is proved in a dedicated section. Planar graphs
Recall that the girth of a graph refers to the length of its shortest cycles. We denote by P g the family of planar graphs having girth at least g . Then, note that P is nothing butthe whole family of planar graphs, while P is the family of triangle-free planar graphs.Towards establishing analogues of the Four-Color Theorem and of Grötzsch’s Theoremfor signed graphs, several works have been dedicated to studying the parameters χ sp ( P g ) and χ s ( P g ) . Note that it is worthwhile investigating such aspects, since, for all values of g ≥ , these two chromatic parameters are known to be finite, due to the existence of an(sp-)bound of P g (see [19]).Let us now discuss the best known bounds on χ sp ( P g ) and χ s ( P g ) for small valuesof g . Regarding the whole family P of planar graphs, it is known that ≤ χ sp ( P ) ≤ and ≤ χ s ( P ) ≤ hold, as proved in [1] and [19], respectively. In particular, it isworth mentioning that if χ s ( P ) = 10 , then there even exists a bound of P of order .Ochem, Pinlou and Sen have actually shown in [19] that if such a bound exists, it mustbe isomorphic to ( SP +9 , (cid:3) + ) , a signed graph we describe in upcoming Section 2. Due toTheorems 1.4 and 1.6, one may equivalently express this result in the following fashion.4 heorem 1.7 (Ochem, Pinlou, Sen [19]) . If there is a double switching sp-bound of P oforder , then it is sp-isomorphic to ( ˆ SP +9 , ˆ (cid:3) + ) . Our first main result in this work (proved in Section 3) is that Theorem 1.7 can bestrengthened, in the sense that it also holds when dropping the double switching require-ment from the statement. That is, we show that the only possible minimal sp-bound of P of order has to be ( ˆ SP +9 , ˆ (cid:3) + ) . Theorem 1.8.
If there is a minimal sp-bound of P of order , then it is sp-isomorphicto ( ˆ SP +9 , ˆ (cid:3) + ) . It is worth mentioning that, supported by computer experimentations and theoreticalevidences, it is conjectured that ( SP +9 , (cid:3) + ) is indeed a bound of P (see [5]). Triangle-free planar graphs
For the family P of triangle-free planar graphs, it is known that ≤ χ sp ( P ) ≤ and ≤ χ s ( P ) ≤ hold, as proved in [19]. A natural intuition is that if χ s ( P ) = 10 indeed held, then it would not be too surprising to have χ s ( P ) = 6 . In practice, however,making that step would not be that easy, as, in general, bounds are seemingly difficult toprove. From that point of view, it would be interesting to have an analogous version ofTheorem 1.8 in this context. Our second main result in this paper (proved in Section 4) liesin that spirit, and reads as follows (where, again, the description of ( SP +5 , (cid:3) + ) , a signedgraphs of order , is postponed to Section 2). Theorem 1.9.
If there is a bound of P of order , then it is isomorphic to ( SP +5 , (cid:3) + ) . K n -minor-free graphs Let F n denote the family of K n -minor-free graphs. It is known that χ sp ( F n ) = 1 , , [1]and χ s ( F n ) = 1 , , for n = 2 , , , respectively [15]. In [5], it was shown that if χ s ( P ) = 10 (which would imply χ sp ( P ) = 20 ) held, then it would imply χ s ( F ) = 10 (and thus χ sp ( F ) = 20 as well). However, prior to studying (sp-)bounds of F n , a first significantstep could be to first investigate analogues of Hadwiger’s Conjecture. To progress towardssuch analogues, it is first important to investigate what types of lower and upper bounds of χ sp ( F n ) and χ s ( F n ) one can expect. In particular, are χ sp ( F n ) and χ s ( F n ) upper boundedat all? In this work, our third main result (proved in Section 5) is the following series ofresults towards those concerns. Theorem 1.10.
The following inequalities hold:(i) For all n ≥ , χ s ( F n ) ≤ (cid:18) n − (cid:19) ( n − ) − and χ sp ( F n ) ≤ (cid:18) n − (cid:19) ( n − ) − . (ii) For all n ≥ , χ sp ( F n ) ≥ ( n +1 − , when n is even , n +1 − , when n is odd . (iii) For all n ≥ , χ s ( F n ) ≥ ( n − , when n is even , n − , when n is odd . raphs with given maximum degree Let G c∆ denote the family of connected graphs with maximum degree at most ∆ . Forlarge values of ∆ , it is possible to mimic an existing proof from [3] (related to the pushablechromatic number of oriented graphs ) to show the following result. Theorem 1.11.
For all ∆ ≥ , we have ∆2 − ≤ χ s ( G c∆ ) ≤ (∆ − · (∆ − · ∆ − + 2 . Since this result can be established by following the exact same lines as the prooffrom [3], there would be no point giving a proof, and we instead refer the reader to thatreference.For smaller values of ∆ , one natural question is whether one can come up with theexact value of χ s ( G c∆ ) . It is worth mentioning that the oriented analogue of this veryquestion remains unanswered, even for the smallest values of ∆ (see [3]). In the case ofsigned graphs, we answer that question for the case ∆ = 3 , which stands as our fourthmain result (proved in Section 6) in this paper. Theorem 1.12.
We have χ s ( G c3 ) = 6 . In fact, we will prove the following stronger result that implies Theorem 1.12 as acorollary. In the statement, recall that ( SP +5 , (cid:3) + ) refers to a signed graph that will bedefined in Section 2; the only important thing to know, at this point, is that it has order . Theorem 1.13.
Every signed subcubic graph with no connected component isomorphic to ( K , ∅ ) or ( K , E ( K )) admits a homomorphism to ( SP +5 , (cid:3) + ) .
2. Definitions, terminology, and preliminary results on Paley graphs
Perhaps one of the most challenging aspects of studying (sp-)homomorphisms of signedgraphs is to exhibit (sp-)bounds. In what follows, we introduce a few popular such boundsthat appeared in the literature, which are related to so-called
Paley graphs .Let q ≡ be a prime power, and F q be the finite field of order q . The signedPaley graph ( SP q , (cid:3) ) of order q is the signed graph with set of vertices V ( SP q ) = F q ,set of positive edges (cid:3) − (+) = { uv : u − v is a square in F q } , and set of negative edges (cid:3) − ( − ) = { uv : u − v is not a square in F q } . The signed Paley plus graph ( SP + q , (cid:3) + ) oforder q + 1 is the signed graph obtained from ( SP q , (cid:3) ) by adding a vertex ∞ and makingit adjacent to every other vertex through a positive edge. To avoid ambiguities, we willrefer to a vertex i = ∞ of SP q or SP + q by writing i . See Figure 1 for an illustration.Signed Paley graphs, signed Paley plus graphs, and their respective double switchinggraphs are, in the literature, regularly used as (sp-)bounds. One reason for that is that thesegraphs have a very symmetric structure, resulting in properties that are very useful when itcomes to designing homomorphisms. Such useful properties deal, in particular, with some Without entering too much into the details, the reader should be aware of a parallel line of researchdedicated to the so-called oriented chromatic number and pushable chromatic number of oriented graphs,which are, roughly speaking, a counterpart of the sign-preserving chromatic number and the signed chro-matic number of signed graphs in which edges are oriented instead of signed. Although the studies of thesigned chromatic number and of the oriented chromatic number are sometimes quite comparable, thereexist contexts in which they actually differ significantly. For instance, there exist undirected graphs withoriented chromatic number arbitrarily larger than their signed chromatic number, as well as undirectedgraphs with signed chromatic number arbitrarily larger than their oriented chromatic number [4]. ∞ (a) ∞ (b) (c)Figure 1: The signed graph ( SP +5 , (cid:3) + ) (a), the signed graph ( SP +5 , (cid:26) + ) obtained by switching the vertices ∞ and of ( SP +5 , (cid:3) + ) (b), and the signed graph ( SP , (cid:3) ) (c). In (a), (b) and (c), solid edges are positiveedges. In (a) and (b), dashed edges are negative edges. In (c), non-edges are negative edges. particular notions of transitivity. More precisely, a signed graph ( G, σ ) is sign-preservingvertex-transitive (or sp-vertex-transitive for short) if, for every two vertices u, v ∈ V ( G ) ,there exists an sp-isomorphism f of ( G, σ ) to itself such that f ( u ) = v . Furthermore, ( G, σ ) is sign-preserving edge-transitive (or sp-edge-transitive for short) if, for every twoedges uv, u ′ v ′ ∈ E ( G ) with the same sign, there exists an sp-isomorphism f of ( G, σ ) to itself such that f ( u ) = u ′ and f ( v ) = v ′ . Similarly, ( G, σ ) is vertex-transitive if, forevery two vertices u, v ∈ V ( G ) , there exists an isomorphism f of ( G, σ ) to itself such that f ( u ) = v ; while ( G, σ ) is edge-transitive if, for every two edges uv, u ′ v ′ ∈ E ( G ) , there existsan isomorphism f of ( G, σ ) to itself such that f ( u ) = u ′ and f ( v ) = v ′ . Proposition 2.1 (Ochem, Pinlou, Sen [19]) . Let q ≡ be a prime power. Then:(i) ( SP q , (cid:3) ) is sp-vertex-transitive and sp-edge-transitive;(ii) ( SP + q , (cid:3) + ) is vertex-transitive and edge-transitive. Given a positive edge uv of a signed graph ( G, σ ) , we call u a positive neighbor of v .Analogously, u is a negative neighbor of v if uv is a negative edge. We denote by N ( v ) , N + ( v ) and N − ( v ) the sets of neighbors, positive neighbors, and negative neighbors, re-spectively, of v in ( G, σ ) . Analogously, we define the degree d ( v ) , positive degree d + ( v ) ,and negative degree d − ( v ) of v as | N ( v ) | , | N + ( v ) | and | N − ( v ) | , respectively. Assuming u and v are two distinct vertices having a common neighbor w , we say that u and v agree on w if w ∈ N α ( u ) ∩ N α ( v ) for some α ∈ {− , + } . Conversely, we say that u and v disagreeon w if they do not agree on w .Let ~v = ( v , . . . , v k ) be a k -tuple of distinct vertices of ( G, σ ) and let ~α = ( α , . . . , α k ) ∈{ + , −} k be a k -vector with each of its elements being + or − . We define the ~α -neighborhood of ~v as N ~α ( ~v ) = ∩ ki =1 N α i ( v i ) . Moreover, we say that ( G, σ ) has property P k,ℓ if, for every k -tuple ~v and every k -vector ~α , we have | N ~α ( ~v ) | ≥ ℓ . We also define the negation − ~α of a k -vector ~α = ( α , . . . , α k ) as − ~α = ( − α , . . . , − α k ) where − α i = − if α i = + , and − α i = + otherwise. The switched ~α -neighborhood of ~v is then ˆ N ~α ( ~v ) = N ~α ( ~v ) ∪ N − ~α ( ~v ) . Lastly, we say that ( G, σ ) has property ˆ P k,ℓ if, for every k -tuple ~v and every k -vector ~α , wehave | ˆ N ~α ( ~v ) | ≥ ℓ . Notice that this property is invariant under the switching operation.7t turns out that signed Paley graphs and signed Paley plus graphs also have the follow-ing interesting properties, which are very convenient ones for designing homomorphisms. Proposition 2.2 (Ochem, Pinlou, Sen [19]) . Let q ≡ be a prime power. Then:(i) ( SP q , (cid:3) ) has property P , q − and P , q − ;(ii) ( SP + q , (cid:3) + ) has property ˆ P ,q , ˆ P , q − and ˆ P , q − .
3. Proof of Theorem 1.8
Let ( T, λ ) be a minimal sp-bound of P of order , assuming that such a signed graphexists. In this section, our goal is to show that ( T, λ ) must be sp-isomorphic to ( ˆ SP +9 , ˆ (cid:3) + ) .To this end, we first use the following lemma to show that ∆( T ) ∈ { , } . We then dealwith each of the two possible values of ∆( T ) separately. Lemma 3.1.
For every vertex v of ( T, λ ) , we have d + ( v ) , d − ( v ) ≥ . Moreover, if d α ( v ) =9 for an α ∈ { + , −} , then the induced subgraph ( T, λ )[ N α ( v )] is sp-isomorphic to ( SP , (cid:3) ) .Proof. It is known (see [12]) that, for the family O of outerplanar graphs, we have χ sp ( O ) = 9 and the only sp-bound of O of order is ( SP , (cid:3) ) . Thus, there existsan outerplanar signed graph ( O, ϕ ) with χ sp (( O, ϕ )) = 9 , and such that the only signedgraph of order to which ( O, ϕ ) admits an sp-homomorphism is ( SP , (cid:3) ) . Also, it is knownfrom [1] that there exists a planar signed graph ( P, π ) with χ sp (( P, π )) = 20 .Let us now consider the planar signed graph ( P ′ , π ′ ) obtained as follows: start from ( P, π ) , and, for every v ∈ V ( P ) , add a copy of ( O, ϕ ) to the + -neighborhood of v andanother copy to the − -neighborhood of v (see Figure 2). v v ( O, ϕ ) (
O, ϕ ) Figure 2: Construction of ( P ′ , π ′ ) in Lemma 3.1. Solid edges are positive edges, dashed ones are negative. Observe that the so-obtained signed graph ( P ′ , π ′ ) is planar. Also, according to ourassumption, ( P ′ , π ′ ) sp −→ ( T, λ ) . Therefore, for each α ∈ { + , −} , we obtain d αT ( v ) > χ sp (( P ′ [ N α ( v )] , π ′ )) = χ sp (( O, ϕ )) = 9 . The last part of the statement follows from the fact that ( SP , (cid:3) ) is the only signed graphof order to which ( O, ϕ ) admits an sp-homomorphism.From the previous result, we deduce that the maximum degree ∆( T ) of T is or .We first consider the case ∆( T ) = 18 , and show that ( T, λ ) is sp-isomorphic to ( ˆ SP +9 , (cid:3) + ) .Note that, by Theorem 1.7, we just have to show that ( T, λ ) is a double switching graph. Lemma 3.2. If ∆( T ) = 18 , then ( T, λ ) is a double switching graph. roof. Observe that Lemma 3.1 ensures that δ ( T ) = 18 . Therefore, if ∆( T ) = 18 , then T is -regular and thus has an anti-matching. Let now v and v ′ be two non-adjacentvertices of T , and assume that v and v ′ are not anti-twins. Then there exists a vertex w ∈ N α ( v ) ∩ N α ( v ′ ) for some α ∈ { + , −} . Observe now that N α ( w ) contains both v and v ′ , and hence induces a non-complete graph. This is a contradiction with Lemma 3.1since N α ( w ) induces ( SP , (cid:3) ) whose underlying graph is complete. Therefore, every pairof non-adjacent vertices of ( T, λ ) are anti-twins, implying that ( T, λ ) is a double switchinggraph.This concludes the proof of Theorem 1.8 in the case ∆( T ) = 18 . The rest of this sectionis devoted to the case ∆( T ) = 19 , in which we aim for a contradiction. To obtain thiscontradiction, we investigate how the neighborhoods of adjacent vertices interact in ( T, λ ) . Lemma 3.3.
For every edge uv of ( T, λ ) and α, β ∈ { + , −} , we have | N α ( u ) ∩ N β ( v ) | ≥ .Proof. As mentioned earlier, there exist planar signed graphs ( P, π ) with χ sp (( P, π )) = 20 ,and, because ( T, λ ) is minimal, that have the following property: for every edge uv ∈ E ( T ) and for any sp-homomorphism f : ( P, π ) sp −→ ( T, λ ) , there exists an edge xy ∈ E ( P ) suchthat f ( x ) = u and f ( y ) = v .Let ( P , M ) denote the signed path on five edges whose three negative edges induce amaximum matching M . Observe that χ sp (( P , M )) = 4 .Let us now consider the planar signed graph ( P ′ , π ′ ) obtained as follows (see Figure 3):start from ( P, π ) , and, for every xy ∈ E ( P ) and all ( α, β ) ∈ { + , −} , include a copyof ( P , M ) inside N α ( x ) ∩ N β ( y ) . Observe that the so-obtained signed graph ( P ′ , π ′ ) is x y x y ( P , M )( P , M )( P , M )( P , M ) Figure 3: Construction of ( P ′ , π ′ ) in Lemma 3.3. Solid edges are positive edges, dashed ones are negative. planar. Furthermore, according to our assumption, ( P ′ , π ′ ) sp −→ ( T, λ ) . Therefore, for every uv ∈ E ( T ) and for all α, β ∈ { + , −} , every copy of ( P , M ) must admit a homomorphismto the subgraph of ( T, λ ) induced by N α ( u ) ∩ N β ( v ) . Hence, the fact that χ sp (( P , M )) = 4 implies that | N α ( u ) ∩ N β ( v ) | ≥ .In view of Lemmas 3.1 and 3.3, every intersection N α ( u ) ∩ N β ( v ) induces a completesubgraph of ( SP , (cid:3) ) of order at least . Before completing the proof, we investigate thepossible signatures of the K ’s that are subgraphs of ( SP , (cid:3) ) , and state some of theirproperties. Since these properties are easy to verify due to the vertex-transitivity andedge-transitivity of ( SP , (cid:3) ) , some formal proofs are omitted.Let ( K , M − ) be the signed graph having the complete graph K as its underlyinggraph and a perfect matching as its set of negative edges. Similarly, let ( K , M + ) be thesigned graph having K as its underlying graph and the edges of a -cycle (that is, thecomplement of a perfect matching) as its set of negative edges.9 bservation 3.4. For every vertex v of ( SP , (cid:3) ) , the set N + ( v ) induces ( K , M + ) in ( SP , (cid:3) ) , while the set N − ( v ) induces ( K , M − ) . Observation 3.5.
For every induced ( K , M + ) (resp., ( K , M − ) ) of ( SP , (cid:3) ) , there existsa v ∈ V ( SP ) such that N + ( v ) (resp., N − ( v ) ) induces that ( K , M + ) (resp., ( K , M − ) ). We are now ready to derive the desired contradiction in the case ∆( T ) = 19 . We firstneed to show that there are two vertices u and v of “large” degree. Lemma 3.6.
For some { α, α } = { + , −} , there exists an α -edge uv of ( T, λ ) such that d α ( u ) = d α ( v ) = 10 and d α ( v ) = 9 .Proof. Since ∆( T ) = 19 , there is a vertex v ∈ V ( T ) with d ( v ) = 19 . By Lemma 3.1, wehave d α ( v ) = 9 and d α ( v ) = 10 for some { α, α } = { + , −} . By Lemma 3.3, each vertex in N α ( v ) has at least four α -neighbors in N α ( v ) . Hence there are at least 40 α -edges between N α ( v ) and N α ( v ) in ( T, λ ) . Since d α ( v ) = 9 , there exists u ∈ N α ( v ) incident to at least ⌈ / ⌉ = 5 such α -edges. Moreover, since d α ( v ) = 9 , Lemma 3.1 ensures that N α ( v ) induces ( SP , (cid:3) ) , and hence u has four α -neighbors in N α ( v ) . Observe also that v is an α -neighbor of u . Thus, we deduce that d α ( u ) ≥ as desired.Let uv be an α -edge of ( T, λ ) that is as described in Lemma 3.6. We now exhibitproperties of the neighborhoods of u and v in ( T, λ ) . Lemma 3.7.
Let A = N α ( v ) ∩ N α ( u ) . We have | A | = 4 . Moreover, there exists x ∈ N α ( u ) such that A = N α ( u ) ∩ N α ( x ) .Proof. Recall that the signed subgraphs induced by N α ( v ) and N α ( u ) are both isomorphicto ( SP , (cid:3) ) , due to Lemma 3.1. Observe that A coincides with the α -neighborhood of u in N α ( v ) . In particular, since u ∈ N α ( v ) , A contains exactly four vertices inducing ( K , M α ) according to Observation 3.4. Furthermore, by Observation 3.5, because A induces ( K , M α ) inside N α ( u ) (which is also sp-isomorphic to ( SP , (cid:3) ) ), there exists x ∈ N α ( u ) such that A ⊆ N α ( x ) . Observe that A is precisely the α -neighborhood of x inthe subgraph induced by N α ( u ) . Hence, A = N α ( x ) ∩ N α ( u ) .We now reach a contradiction by showing that N α ( x ) has size 9 (and thus induces ( SP , (cid:3) ) ) and contains two disjoint copies of ( K , M α ) , which is impossible. These state-ments are summarized in the following lemmas. Lemma 3.8.
The sets B = N α ( u ) \ ( A ∪ { x } ) and C = N α ( v ) ∩ N α ( u ) are disjoint andthey both induce ( K , M α ) in ( T, λ ) . Moreover, we have B = N α ( x ) ∩ N α ( u ) .Proof. Since N α ( u ) induces the signed complete graph SP , B is the set of all neighborsof x in N α ( u ) that are not in A , i.e., all the α -neighbors of x . Hence, B = N α ( x ) ∩ N α ( u ) .Observation 3.4 thus yields that B induces ( K , M α ) .The same argument, applied to the copy of SP induced by N α ( v ) , ensures that C alsoinduces ( K , M α ) . Moreover, since B ⊂ N α ( u ) and C ⊂ N α ( u ) , these sets are disjoint. Lemma 3.9. N α ( x ) contains B ∪ C and has size 9.Proof. First observe that, by Lemma 3.8, the vertices of B are α -neighbors of x . Hence, B ⊂ N α ( x ) . Now, since v has degree 19, v and x are adjacent and Lemma 3.3 ensures that N α ( v ) contains at least four α -neighbors of x . Observe now that N α ( v ) = A ∪ C ∪ { u } and that A ∪ { u } are α -neighbors of x (by Lemma 3.7). Therefore, the four α -neighborsof x in N α ( v ) are precisely the vertices of C , i.e. C = N α ( v ) ∩ N α ( x ) ⊂ N α ( x ) .10e now exhibit 10 α -neighbors of x , which will ensure that | N α ( x ) | = 9 . By Lemma 3.7,we already know five such neighbors, namely u and the vertices in A . Moreover, since N α ( v ) = A ∪ C ∪ { u } and C ⊂ N α ( x ) , we get that x / ∈ N α ( v ) , and, hence, v is another α -neighbor of x . Now, by Lemma 3.3, there are at least four α -neighbors of x in N α ( v ) .Thus, because x has four more α -neighbors in A and two more in { u, v } , x has a total of10 α -neighbors. So, we finally deduce that | N α ( x ) | = 10 , and, thus, that | N α ( x ) | = 9 .
4. Proof of Theorem 1.9
Let ( T, λ ) be a minimal bound of P of order . We will show that ( T, λ ) must beisomorphic to ( SP +5 , (cid:3) + ) by essentially proving that ( T, λ ) must have very specific proper-ties, converging towards the precise ones that ( SP +5 , (cid:3) + ) has. To do so, we will constructsome signed graphs ( H , π ) , ( H , π ) , . . . , all being triangle-free and planar, and, thus,admitting homomorphisms to ( T, λ ) . These ( H i , π i ) ’s will be constructed gradually, so thateach of the ( H i , π i ) ’s allows to deduce more properties of ( T, λ ) .To construct these ( H i , π i ) ’s, we will mainly use the triangle-free planar signed graph ( H, π ) depicted in Figure 4 as a building block. In what follows, it is important to keep inmind that we deal with the vertices and edges of ( H, π ) using the notation introduced inFigure 4. xya x,y a x,y a x,y d x,y d x,y d x,y Figure 4: The main gadget ( H, π ) used to prove Theorem 1.9. Solid edges are positive edges. Dashed edgesare negative edges. Vertices x and y agree on a x,y , a x,y and disagree on d x,y , d x,y . Given a signed graph ( G, Σ) and one of its vertices v , by pinning ( H, π ) on v we meanstarting from ( G, Σ) , adding a copy of ( H, π ) , and identifying the vertex x of ( H, π ) withthe vertex v of ( G, Σ) . Similarly, for two distinct vertices u and v of ( G, Σ) , by pinning ( H, π ) on ( u, v ) we mean starting from ( G, Σ) , adding a copy of ( H, π ) , identifying thevertex x of ( H, π ) with the vertex u of ( G, Σ) , and similarly identifying y with v . Observethat if ( G, Σ) is a triangle-free planar signed graph, and u and v are two non-adjacentvertices of ( G, Σ) belonging to a same face, then the signed graph obtained from ( G, Σ) bypinning ( H, π ) on ( u, v ) is also a triangle-free planar signed graph.Note that the vertices of ( H, π ) are named as functions of x and y . This will allow us torefer to vertices of a copy of ( H, π ) after pinning it to, say, ( u, v ) of ( G, Σ) , as functions of u and v . Since we will deal with larger and larger signed graphs containing multiple copiesof ( H, π ) , this terminology will allow us to refer to particular vertices in an unambiguousway.We first show that ( T, λ ) must be a signed complete graph. This is done by makinguse of the following observation. Recall that a negative cycle in a signed graph is a cyclehaving an odd number of negative edges, while a positive cycle has an even number ofnegative edges. 11 bservation 4.1 (see e.g. [15], Lemma 3.10) . Two vertices of a signed graph have distinctimages under every homomorphism if and only if they are adjacent or they are part of anegative -cycle. In the next result, we construct some first triangle-free planar signed graphs, from whichwe get that ( T, λ ) must indeed be complete. We start from ( H , π ) being the signed graph ( H, π ) itself, in which we slightly modify the names of the vertices. That is, we refer tothe vertices of ( H , π ) as in ( H, π ) , except that we omit the superscripts (if any). Thus,the vertices x, y retain their name, while the vertices a x,y , a x,y , a x,y , d x,y , d x,y and d x,y arenow, in ( H , π ) , named a , a , a , d , d and d , respectively. Lemma 4.2. ( T, λ ) is a signed complete graph.Proof. Let ( H , π ) be the signed graph obtained from ( H , π ) by pinning ( H, π ) on ( x, a ) .Note that ( H , π ) is a triangle-free planar signed graph, and, thus, according to ourassumption there exists a homomorphism g : ( H , π ) → ( T, λ ) . By Observation 4.1, thevertices x , y , a , a , d and d of ( H , π ) have distinct images in ( T, λ ) under everyhomomorphism ( H , π ) → ( T, λ ) . Furthermore, observe that the images of the vertices x , a , a x,a , a x,a , d x,a and d x,a are also distinct. Therefore, since x , y and a must have distinctimages and ( T, λ ) has exactly six vertices, the images of a x,a , a x,a , d x,a and d x,a must containthe image of y . In other words, we must have g ( y ) ∈ { g ( a x,a ) , g ( a x,a ) , g ( d x,a ) , g ( d x,a ) } .Therefore, g ( x ) must be adjacent to { g ( a ) , g ( a ) , g ( d ) , g ( d ) , g ( y ) } in ( T, λ ) , hence hasdegree 5.Next, let ( H , π ) be the signed graph obtained in the following manner: for eachvertex v of ( H , π ) , we glue a copy of ( H , π ) by identifying the vertex x of ( H , π ) withthe vertex v of ( H , π ) . Note that ( H , π ) is also a triangle-free planar signed graph.Therefore, it admits a homomorphism to ( T, λ ) . By a previous remark, the vertices x , y , a , a , d and d must have distinct images by every homomorphism ( H , π ) → ( T, λ ) . Hence,there must be six distinct vertices of degree in ( T, λ ) , which thus must be complete.Let now ( H , π ) be the signed graph obtained from ( H , π ) by pinning four ( H, π ) ’son ( x, a ) , ( x, d ) , ( y, a ) and ( y, d ) , respectively. Note that ( H , π ) is a triangle-free planarsigned graph, and thus it admits a homomorphism to ( T, λ ) . In what follows, we need tounderstand better the different types of homomorphisms of ( H , π ) to ( T, λ ) .Let f be a homomorphism of ( H , π ) to ( T, λ ) . For convenience, suppose that V ( T ) = { , , , , , } . By Observation 4.1, we know that, in ( H , π ) , the vertices x , y , a , a , d and d have distinct images by f . Without loss of generality, we may assume that theseimages by f are as displayed in the following table: f ( x ) f ( y ) f ( a ) f ( a ) f ( d ) f ( d )1 2 3 4 5 6 Furthermore, by Observation 4.1 we know that f ( a ) ∈ { , } and f ( d ) ∈ { , } . Thus,without loss of generality, we may also assume f ( a ) = 5 , which implies f ( d ) = 3 : f ( a ) f ( d )5 3 Note that this may require to switch some vertices among a and d , but in that case we canrelabel some vertices of the four pinned copies of ( H, π ) in ( H , π ) and keep the originalsignature of ( H , π ) . 12ow, let us focus on the copy of ( H, π ) in ( H , π ) that was pinned on ( x, a ) . No-tice, by Observation 4.1, that f ( x ) , f ( a ) , f ( a x,a ) , f ( a x,a ) , f ( d x,a ) and f ( d x,a ) are pairwisedistinct, that f ( a x,a ) , f ( a x,a ) ∈ { , , } (since they agree on vertices and ), and that f ( d x,a ) , f ( d x,a ) ∈ { , , } (since they disagree on vertices and ). Therefore, either f ( a x,a ) = 3 or f ( a x,a ) = 3 and, similarly, either f ( d x,a ) = 4 or f ( d x,a ) = 4 . As we havenot assumed that ( H , π ) is embedded in the plane in a specific way, due to the sym-metric structure of the graph, we may assume without loss of generality f ( a x,a ) = 3 and f ( d x,a ) = 4 . Reasoning similarly on the other copies of ( H, π ) , we may suppose that wehave the following images by f : f ( a x,a ) f ( d x,a ) f ( a x,d ) f ( d x,d )3 4 5 6 f ( a y,a ) f ( d y,a ) f ( a y,d ) f ( d y,d )3 4 6 5 We now analyze the possible images by f for some of the remaining vertices of ( H , π ) .First, note that, by Observation 4.1, we have the following: Observation 4.3.
By Observation 4.1, we have: • { f ( a y,a ) , f ( d y,a ) } = { , } , • { f ( a x,d ) , f ( d x,d ) } = { , } , • { f ( a y,d ) , f ( d y,d ) } = { , } , • { f ( a x,a ) , f ( d x,a ) } = { , } . Regarding the first item in Observation 4.3, there are two possibilities for f , namelyeither ( f ( a y,a ) , f ( d y,a )) = (1 , , or conversely ( f ( a y,a ) , f ( d y,a )) = (6 , . In the next twolemmas, we analyze the consequences on f of being in one case or the other. Lemma 4.4. If f ( a y,a ) = 1 and f ( d y,a ) = 6 , then we have the following images by f : f ( a x,a ) f ( d x,a ) f ( a x,d ) f ( d x,d )6 2 2 4 f ( a y,a ) f ( d y,a ) f ( a y,d ) f ( d y,d )1 6 1 4 Proof. If f ( d x,d ) = 2 , then the positive cycle a y,a ya y,a aa y,a and the negative cycle xd x,d da x,d x of ( H , π ) have the same image by f , which is a contradiction. Therefore, f ( a x,d ) =2 and f ( d x,d ) = 4 . Also, if f ( a y,d ) = 4 , then the negative cycle xd x,d da y,d ya x has image by f , which is a positive closed walk in ( T, λ ) , a contradiction. From this, wededuce that f ( a y,d ) = 1 and f ( d y,d ) = 4 . Finally, if f ( a x,a ) = 2 , then the positive cy-cle xa x,a aa x,a x and the negative cycle a y,d yd y,d da y,d have the same image by f , acontradiction. Therefore, f ( a x,a ) = 6 and f ( d x,a ) = 2 . Lemma 4.5. If f ( a y,a ) = 6 and f ( d y,a ) = 1 , then we have the following images by f : ( a x,a ) f ( d x,a ) f ( a x,d ) f ( d x,d )2 6 4 2 f ( a y,a ) f ( d y,a ) f ( a y,d ) f ( d y,d )6 1 4 1 Proof. If f ( a x,d ) = 2 , then the positive cycle xa x,d da x,d x and the negative cycle d y,a ya y,a ad y,a of ( H , π ) have the same image by f , which is not possible. Therefore, f ( a x,d ) = 4 and f ( d x,d ) = 2 . Now, if f ( d y,d ) = 4 , then the negative cycle xa x,d dd y,d ya x has im-age by f , which is a positive closed walk in ( T, λ ) , a contradiction from whichwe deduce f ( a y,d ) = 4 and f ( d y,d ) = 1 . Similarly, if f ( a x,a ) = 6 , then the negative cy-cle xd ya y,a aa x,a x has image by f , which is a positive closed walk in ( T, λ ) , acontradiction. Then we deduce that f ( a x,a ) = 2 and f ( d x,a ) = 6 .From Lemmas 4.4 and 4.5, we get that there are, thus far, two possible partial extensionsfor f . We denote by f the one described in the statement of Lemma 4.4, and by f theone described in the statement of Lemma 4.5.Let { i , . . . , i } = { , . . . , } . In the signed graph ( T, λ ) , if two vertices i , i agree ontwo vertices i , i and disagree on i , i , then we say that { i , i } is a splitter that yieldstwo teams { i , i } and { i , i } . Naturally, in this case, i is in the same team as i , that isopposite to the team of i and i . Observe that no matter how we switch vertices in ( T, λ ) ,the pair { i , i } remains a splitter yielding the same two teams. Upon switching vertices,it may happen that i , i get to disagree on i , i and to agree on i , i – but the fact that { i , i } yields teams { i , i } and { i , i } cannot be lost.Having a closer look, in ( H , π ) , at the images by f , observe that x and y must agreeon a and a and disagree on d and d . The images by f of x and y thus imply that,in ( T, λ ) , the pair { , } is a splitter yielding teams { , } and { , } . Moreover, because f ( a ) = 5 and there is a copy of ( H, π ) pinned to ( x, a ) , then, in ( T, λ ) , the pair { , } mustbe a splitter. Similarly, because f ( d ) = 3 , the pair { , } must also be a splitter. Thus,vertex is part of at least three distinct splitters. We actually need to show somethingstronger. Lemma 4.6.
Every vertex of ( T, λ ) is part of at least four distinct splitters.Proof. Let ( H , π ) be the triangle-free planar signed graph obtained by pinning one copy of ( H, π ) to each of the eight pairs ( a x,a , a x,a ) , ( d x,a , d x,a ) , ( a x,d , a x,d ) , ( d x,d , d x,d ) , ( a y,a , a y,a ) , ( d y,a , d y,a ) , ( a y,d , a y,d ) and ( d y,d , d y,d ) of vertices of ( H , π ) . Consider an extension of f to ( H , π ) . Note that if f is extended so that it matches f , then the copy of ( H, π ) pinnedon ( a y,d , a y,d ) implies that { , } is a splitter. If f is extended so that it matches f , thenthe copy of ( H, π ) pinned on ( d y,a , d y,a ) implies that { , } is a splitter. Earlier, we havealready pointed out that { , } , { , } and { , } are splitters. Therefore, because ( H , π ) verifies f ( x ) = 1 , we get that vertex must be part of at least four splitters.Let now ( H , π ) be the triangle-free planar signed graph obtained by starting from ( H , π ) and, for each of its vertices u , adding a copy of ( H , π ) and identifying u andthe vertex x of that copy. Then, for every homomorphism ( H , π ) → ( T, λ ) and for every i ∈ V ( T ) , there is a copy of ( H , π ) in ( H , π ) for which the image of x is i . Thiscompletes the proof.In what follows, we prove that if some vertex of ( T, λ ) is part of five distinct splitters,then ( T, λ ) must be isomorphic to ( SP +5 , (cid:3) + ) , in which case we are done.14 emma 4.7. If a vertex of ( T, λ ) is part of five distinct splitters, then ( T, λ ) is isomorphicto ( SP +5 , (cid:3) + ) .Proof. Without loss of generality, assume that, in ( T, λ ) , vertex is part of five distinctsplitters. Switch the − -neighbors of vertex so that all its incident edges get positive.Because the set { i, } is a splitter for every i ∈ { , , , , } , we deduce that every vertex i must be incident to exactly two positive edges and two negative edges in ( T − , λ ) , thesigned graph obtained from ( T, λ ) by deleting vertex . Then, the vertices in { , , , , } and their incident positive edges must induce a -regular graph. Since the only -regular(simple) graph of order is the -cycle, we get the desired conclusion.Now assume that no vertex of ( T, λ ) is part of five distinct splitters. We prove that,under that assumption, ( T, λ ) must be one of two possible signed graphs, ( K , M ) and ( K , M ) , defined as follows. Let M be a perfect matching of K , the complete graph oforder . The signed graph ( K , M ) is the signed K in which the set of negative edges isprecisely M . The signed graph ( K , M ) is the signed K in which the set of negative edgesis the set M = E ( K ) \ M of the edges that are not in M . Lemma 4.8.
If no vertex of ( T, λ ) is part of five distinct splitters, then ( T, λ ) is isomorphicto ( K , M ) or ( K , M ) .Proof. In this case, each vertex of ( T, λ ) is part of exactly four distinct splitters. Let usswitch the − -neighbors of vertex to make all its incident edges positive. Let ( T − , λ ) be the signed graph obtained from ( T, λ ) by deleting vertex . Note that the subgraph T + induced by the positive edges of ( T − , λ ) must have exactly four vertices of degree . Bythe Handshaking Lemma, the fifth vertex j must then have even degree, hence has degree or . If j has degree , then T + is the disjoint union of a singleton vertex and a -cycle.In this case, by switching j in ( T, λ ) we get the signed graph ( K , M ) . Now, if j has degree , then T + is the -clique-sum of two -cycles. In this case, by switching vertex and j in ( T, λ ) , we obtain the signed graph ( K , M ) .We complete the proof by showing that it is actually not possible for ( T, λ ) to beisomorphic to ( K , M ) or to ( K , M ) , a contradiction with the previous lemma. Lemma 4.9. ( T, λ ) cannot be isomorphic to ( K , M ) or ( K , M ) .Proof. Note that if ( T, λ ) is isomorphic to ( K , M ) or ( K , M ) , then, for every vertex i of ( T, λ ) , there exists exactly one other vertex j such that { i, j } is not a splitter. Let usconsider the two possibilities, f and f , for f to be extended in ( H , π ) .First, assume that f is partially extended as f . The three sets (of cardinality ) ofvertices of ( T, λ ) that are not splitters are { , } , { , } and { , } . Therefore, if ( T, λ ) is isomorphic to ( K , M ) , then its three negative edges are , and . Analogously,if ( T, λ ) is isomorphic to ( K , M ) , then its three positive edges are , and . Now,looking at the structure of ( K , M ) or ( K , M ) , the splitter { , } yields the two teams { , } and { , } . Let us now look further at the images of the vertices of ( H , π ) by f .We know that f ( x ) = 1 and f ( d ) = 3 . Moreover, we know that x and d agree on a x,d and a x,d and disagree on d x,d and d x,d . Because f ( a x,d ) = 5 , f ( a x,d ) = 2 , f ( d x,d ) = 4 and f ( d x,d ) = 6 , we can conclude that the splitter { , } yields the two teams { , } and { , } ,which is a contradiction. Thus if f is extended as f , then ( T, λ ) must be isomorphic to ( SP +5 , (cid:3) + ) .Second, assume that f is partially extended as f . In this case, the three sets (ofcardinality ) of vertices of ( T, λ ) that are not splitters are { , } , { , } and { , } . This15mplies that the splitter { , } yields the two teams { , } and { , } . However, in ( H , π ) ,we have f ( x ) = 1 , f ( d ) = 3 , f ( a x,d ) = 5 , f ( a x,d ) = 4 , f ( d x,d ) = 2 and f ( d x,d ) = 6 .From these images, we conclude that the splitter { , } yields the two teams { , } and { , } , which is a contradiction. Thus if f is extended as f , then, again, ( T, λ ) must beisomorphic to ( SP +5 , (cid:3) + ) .
5. Proof of Theorem 1.10
We start by proving the upper bounds, which we do by exploiting existing connectionsbetween signed graphs and acyclic colorings. Recall that an acyclic coloring of an undi-rected graph G is a proper vertex-coloring such that the subgraph induced by any twodistinct colors is acyclic, i.e., is a forest. The acyclic chromatic number χ a ( G ) of G is theminimum k such that G admits an acyclic k -coloring. Proof of Theorem 1.10(i).
It was proved in [18] that χ a ( G ) ≤ (cid:0) n − (cid:1) holds for every graph G ∈ F n . Furthermore, given a signed graph ( G, σ ) , it is also known that if χ a ( G ) ≤ k , then χ s (( G, σ )) ≤ k k − (see [19]) and χ sp (( G, σ )) ≤ k k − (see [1]). Combining these boundsyields the desired upper bounds.We say that a family F of graphs is complete if for every finite collection C = { G , . . . , G t } of graphs from F , the graph obtained by taking the disjoint union of all graphs of C alsobelongs to F . Lemma 5.1.
Every complete family F of graphs has an sp-bound of order χ sp ( F ) and abound of order χ s ( F ) .Proof. Suppose F does not have an sp-bound of order n = χ sp ( F ) . Let S be the set of allsignatures of K n . Since F does not have any sp-bound of order n , for each π ∈ S thereexists a ( G π , σ π ) that does not admit an sp-homomorphism to ( K n , π ) . Let ( G, σ ) be thesigned graph containing ( G π , σ π ) as a subgraph for all π ∈ S . That is, ( G, σ ) is the disjointunion of all possible ( G π , σ π ) ’s, where π runs across S . Observe that G ∈ F . Furthermore,note that ( G, σ ) does not admit an sp-homomorphism to ( K n , π ) for any π ∈ S . Thus χ sp (( G, π )) > n , a contradiction.The proof for the existence of a bound of order χ s ( F ) is similar.With Lemma 5.1 on hand, we can now prove the second part of Theorem 1.10. Proof of Theorem 1.10(ii).
We know from [13, 19] that the result holds for n = 1 , , , , .We prove the result for larger values of n by induction. Suppose that the result holds forall n ≤ t where t ≥ is odd. We show that the result holds for n = t + 1 and n = t + 2 .We first prove the result for n = t + 1 . Consider the following construction. Given asigned graph ( G, σ ) , take two disjoint copies ( G , σ ) and ( G , σ ) of ( G, σ ) , add a newvertex ∞ , and make every vertex of ( G , σ ) adjacent to ∞ via a positive edge and everyvertex of ( G , σ ) adjacent to ∞ via a negative edge. We denote the so-obtained signedgraph by ( G ∗ , σ ∗ ) . Observe that χ sp (( G ∗ , σ ∗ )) = 2 χ sp (( G, σ )) + 1 . (1)Indeed, if ( G, σ ) sp −→ ( H, π ) with | V ( H ) | = χ sp (( G, σ )) , then ( G ∗ , σ ∗ ) sp −→ ( H ∗ , π ∗ ) , whichensures that χ sp (( G ∗ , σ ∗ )) ≤ | V ( H ∗ ) | = 2 χ sp (( G, σ )) + 1 . For the reverse inequality,assume that there is an sp-homomorphism f : ( G ∗ , σ ∗ ) sp −→ ( H, π ) . Then one of the copiesof ( G, σ ) in ( G ∗ , σ ∗ ) is mapped by f in H [ N + ( f ( ∞ ))] , and the other in H [ N − ( f ( ∞ ))] .16he inequality then follows from the fact that at least one of these subgraphs has order atmost | V ( H ) \{∞}| = χ sp (( G ∗ ,σ ∗ )) − .Let ( H t , π t ) be a signed graph with χ sp (( H t , π t )) ≥ t +1 − , where H t ∈ F t . Let us set ( H t +1 , π t +1 ) = ( H ∗ t , π ∗ t ) . Note that H t +1 ∈ F t +1 ; therefore, by Equation (1), we have χ sp (( H t +1 , π t +1 )) = 2 χ sp (( H t , π t )) + 1 ≥ · t +1 −
43 + 1= 2 t +2 − ( t +1)+1 − . This example implies the lower bound for the case n = t + 1 .We now prove the result for n = t + 2 . First of all, consider the signed graph ( H t +2 , π t +2 ) = ( H ∗ t +1 , π ∗ t +1 ) . By Equation (1) we have χ sp (( H t +2 , π t +2 )) = 2 χ sp (( H t +1 , π t +1 )) + 1 ≥ · ( t +1)+1 −
53 + 1= 2 t +3 −
10 + 33 = 2 ( t +2)+1 − − . Since H t +2 ∈ F t +2 , the result will hold if we can prove that we cannot have equality in thesecond line of the equation above. Thus, assume the contrary, i.e., χ sp (( H t +1 , π t +1 )) = 2 ( t +1)+1 − and χ sp (( H t , π t )) = 2 t +1 − . Now consider the following construction, similar to the ones depicted on Figures 2and 3. Take ( H t +2 , π t +2 ) and | V ( H t +2 ) | copies of ( H ∗ t +1 , π ∗ t +1 ) . After that, for every vertex v ∈ V ( H t +2 ) , take a copy of ( H ∗ t +1 , π ∗ t +1 ) and identify v with the vertex ∞ . We call theresulting graph ( H ′ t +2 , π ′ t +2 ) . We further enhance this construction as follows. Take thedisjoint union of ( H ′ t +2 , π ′ t +2 ) and of | E ( H ′ t +2 ) | copies of ( H t , π t ) . Then, for every edge e = uv ∈ E ( H ′ t +2 ) and every pair ( α, β ) ∈ { + , −} , take a copy of ( H t , π t ) , make itsvertices adjacent to u through α -edges and to v through β -edges. We denote the resultinggraph by ( H ′′ t +2 , π ′′ t +2 ) .If χ sp ( F t +2 ) < ( t +2)+1 − (contradicting the statement of the result we want to prove),then we must have ( t +2)+1 − − > χ sp ( F t +2 ) > χ sp (( H ′′ t +2 , π ′′ t +2 )) > χ sp (( H t +2 , π t +2 )) = 2 ( t +2)+1 − − . This implies that there exists a signed graph ( T, λ ) of order ( t +2)+1 − − such that ( H ′′ t +2 , π ′′ t +2 ) sp −→ ( T, λ ) . Let f : ( H ′′ t +2 , π ′′ t +2 ) sp −→ ( T, λ ) be an sp-homomorphism. Note that f is surjective, since χ sp (( H ′′ t +2 , π ′′ t +2 )) = ( t +2)+1 − − . As we also have χ sp (( H t +2 , π t +2 )) = ( t +2)+1 − − , we get that the vertices of the original ( H t +2 , π t +2 ) contained in ( H ′′ t +2 , π ′′ t +2 ) as a subgraph also map onto the vertices of ( T, λ ) . From this, we may infer that everyvertex x of ( T, λ ) has a copy of ( H t +1 , π t +1 ) mapped to its α -neighborhood by f for every α ∈ { + , −} . Thus every vertex x of ( T, λ ) must have at least χ sp (( H t +1 , π t +1 )) = ( t +1)+1 − α -neighbors for every α ∈ { + , −} . Since T has exactly ( t +2)+1 − − · ( t +1)+1 −
53 + 1 ( T, λ ) must thus have exactly ( t +1)+1 − α -neighbors for every α ∈ { + , −} . Hence, ( T, λ ) is a complete signed graph. Furthermore, for every edge e = uv ∈ E ( H t +2 ) of the original copy contained in ( H ′′ t +2 , π ′′ t +2 ) as a subgraph and forevery pair ( α, β ) ∈ { + , −} , there is a copy of ( H t , π t ) contained in the subgraph inducedby N α ( u ) ∩ N β ( v ) . Thus, in particular, for any distinct pair of vertices x, y of ( T, λ ) , N α ( x ) ∩ N β ( y ) contains at least χ sp (( H t , π t )) = t +1 − vertices for every ( α, β ) ∈ { + , −} .To reach a contradiction, we count in two ways the number of + -edges between A and B . Let x be a vertex of ( T, λ ) . We already know that the + -neighborhood A of x in ( T, λ ) contains exactly ( t +1)+1 − vertices and the − -neighborhood B of x in ( T, λ ) contains exactly ( t +1)+1 − vertices. Moreover, every vertex y in A has exactly t +1 − α -neighbors in A for every α ∈ { + , −} . Note that y already has one + -neighbor, x , and t +1 − + -neighbors in A . Hence, y must have exactly ( t +1)+1 − − t +1 − − t +1 − -neighbors in B . Thus, there are exactly (2 ( t +1)+1 − t +1 − + -edges between the sets A and B . Similarly, every vertex z in B has exactly t +1 − α -neighbors in A for every α ∈ { + , −} . Note that z already has t +1 − + -neighbors in B . Hence, it must have exactly ( t +1)+1 − − t +1 −
43 = 2 t +1 − -neighbors in B . Thus, there are exactly (2 ( t +1)+1 − t +1 − + -edges between the sets A and B . This is a contradiction with the previous counting, which implies that ( T, λ ) cannot exist. This concludes the proof.This leaves us with proving the very last part of Theorem 1.10. Proof of Theorem 1.10(iii). If n is even, Theorems 1.5 and 1.10(ii) give that χ s (( G, σ )) ≥ χ sp (( G, σ )) ≥ n +1 − . Hence, because χ s (( G, σ )) is an integer, we get χ s (( G, σ )) ≥ (cid:24) n +1 − (cid:25) = (cid:24) n − − (cid:25) = 2 n − since n is even. The case when n is odd is similar.
6. Proof of Theorem 1.13
Throughout this section, we say that each of the two signed graphs ( K , ∅ ) (havingpositive edges only) and ( K , E ( K )) (having negative edges only) is a bad K , while everyother signature of K gives a good K .We first observe that ( SP +5 , (cid:3) + ) contains a copy of each good K . Observation 6.1. If ( K , Σ) is not bad, then ( K , Σ) → ( SP +5 , (cid:3) + ) . roof. Given a signature Σ of K , one can switch some vertices to obtain an equivalentsignature Σ ′ in which some vertex v has its three incident edges being positive. If ( K , Σ) is not bad, then the signed graph obtained by deleting v from ( K , Σ ′ ) does not haveonly positive edges or only negative edges and hence can be found in ( SP , (cid:3) ) . Therefore ( SP +5 , (cid:3) + ) contains ( K , Σ ′ ) as a subgraph, where v is mapped to ∞ .In this section, we want to show that the family of all signed subcubic graphs withno bad K as a connected component, admits a homomorphism to ( SP +5 , (cid:3) + ) . The proofis by contradiction. Suppose there exists a signed subcubic graph with no bad K as aconnected component, that does not admit a homomorphism to ( SP +5 , (cid:3) + ) . We focus on ( G, σ ) , a counterexample that is minimal in terms of order. That is, every signed subcubicgraph with fewer vertices than ( G, σ ) admits a homomorphism to ( SP +5 , (cid:3) + ) . Our goal isto show that ( G, σ ) cannot exist, a contradiction. This is done by investigating propertiesof ( G, σ ) , and considering homomorphisms to ( SP +5 , (cid:3) + ) (depicted in Figure 1(a)).By minimality, we observe that ( G, σ ) is connected. Also, G = K (by Observation 6.1).We start off by showing that ( G, σ ) cannot have cut-vertices. Lemma 6.2. ( G, σ ) is -connected.Proof. Assume that ( G, σ ) has a cut-vertex v . Then, removing v from ( G, σ ) results in atleast two connected components. Assume that ( G , σ ) is one such connected component,and ( G , σ ) is the disjoint union of all the other connected components. Let ( G ′ , σ ′ ) bethe signed graph obtained by putting the vertex v back in ( G , σ ) , and let ( G ′ , σ ′ ) be thesigned graph obtained by putting the vertex v back in ( G , σ ) . Note that none of thesetwo signed graphs is cubic, and, thus, none of them can be a bad K . By minimality of ( G, σ ) , there are f : ( G ′ , σ ′ ) → ( SP +5 , (cid:3) + ) and f : ( G ′ , σ ′ ) → ( SP +5 , (cid:3) + ) . Due to thevertex-transitivity of ( SP +5 , (cid:3) + ) , we may assume f ( v ) = f ( v ) . Now, combining f and f yields a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) , a contradiction.Through the next result, we aim at reducing ( G, σ ) to a cubic graph. Note that ( G, σ ) has no vertex of degree 1 since it is 2-connected. Lemma 6.3. ( G, σ ) does not contain a vertex of degree .Proof. Suppose the contrary, i.e., assume that ( G, σ ) contains a degree- vertex u withneighbors v and w . Let ( G ′ , σ ′ ) be the signed graph obtained from ( G, σ ) by deleting u and adding the edge vw (if it was not already present).Observe that if vw was already present in ( G, σ ) , then it is not possible for ( G ′ , σ ′ ) tobe isomorphic to a bad K since v and w have now degree 2 in G ′ . In case we do add theedge vw , we choose its sign in such a way we do not create any bad K . This means that if G ′ is isomorphic to K , then we choose the sign of vw so that one of the -cycles of ( G ′ , σ ′ ) becomes negative. Otherwise, we assign any sign to vw in ( G ′ , σ ′ ) .In all cases, ( G ′ , σ ′ ) cannot be a bad K , and, hence, by minimality of ( G, σ ) , there exists f : ( G ′ , σ ′ ) → ( SP +5 , (cid:3) + ) . Because vw is an edge, we know that f ( v ) = f ( w ) . Note thatthis f also stands as a homomorphism of ( G − u, σ ) to ( SP +5 , (cid:3) + ) . Now, since ( SP +5 , (cid:3) + ) has property ˆ P , according to Proposition 2.2, we can extend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) , a contradiction.Thus, from now on we can assume that ( G, σ ) is cubic. To finish off the proof, weprove that G cannot contain any of the configurations depicted in Figure 5. Throughoutthe rest of this section, whenever dealing with one of these configurations, we do so by19 v v v v v (a) v v v v v v (b) v v v v v v v v (c) v v v v v v v v v v v v v v (d)Figure 5: Configurations reduced for proving Theorem 1.13. Black vertices are vertices having their wholeneighborhood being part of the configuration. White vertices may have neighbors not depicted in theconfiguration. employing the terminology given in the figure. It is important to emphasize that, in theseconfigurations, white vertices are vertices that can have neighbors outside the configuration,while the whole neighborhood of the black vertices is as displayed in the configuration. Inparticular, some of the white vertices could be the same vertices, or be adjacent to eachother.We proceed with the configuration depicted in Figure 5(a). Lemma 6.4. ( G, σ ) does not contain the configuration depicted in Figure 5(a).Proof. Suppose the contrary, i.e., assume that ( G, σ ) contains the configuration depicted inFigure 5(a). Let ( G ′ , σ ′ ) be the signed graph obtained from ( G, σ ) by deleting the vertices v , v , v and v and adding the edge v v (if it was not already present). In case v v doesnot exist in ( G, σ ) , then, in ( G ′ , σ ′ ) , just as in the proof of Lemma 6.3, we choose the signof v v so that ( G ′ , σ ′ ) is not a bad K . Thus, by minimality there exists a homomorphism f : ( G ′ , σ ) → ( SP +5 , (cid:3) + ) . Because ( SP +5 , (cid:3) + ) is edge-transitive, without loss of generalitywe may assume f ( v ) = ∞ and f ( v ) = 1 . Besides, if needed, we can switch some verticesof ( G, σ ) to ensure σ ( v v ) = σ ( v v ) = σ ( v v ) = σ ( v v ) = + . More precisely, we first switch v if σ ( v v ) = − , then switch v if σ ( v v ) = − , then switch v if σ ( v v ) = − , and finally switch v in case σ ( v v ) = − .We first set f ( v ) = ∞ . We now choose i, j, k ∈ V ( SP +5 ) \ {∞} so that i is a σ ( v v ) -neighbor of f ( v ) = 1 , j is a σ ( v v ) -neighbor of i , and k is a σ ( v v ) -neighbor of j . Now,20etting f ( v ) = i , f ( v ) = j and f ( v ) = k , extends f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) , a contradiction.Before proceeding with the next configuration, we first need to state a useful observationthat deals with signatures ( P , σ ) of the -path P = u u u u . Observation 6.5.
Let g be a partial function of V ( P ) to V ( SP ) where only u and u getan image by g . Assume g ( u ) = i and g ( u ) = j for some i, j ∈ V ( SP ) . Then, regardlessof i and j , it is possible to extend g to an sp-homomorphism of ( P , σ ) to ( SP , (cid:3) ) unless σ ( u u ) = σ ( u u ) = σ ( u u ) and i = j .Proof. Due to the transitivity properties of ( SP , (cid:3) ) , it is sufficient to focus on the caseswhere g ( u ) = 1 and g ( u ) ∈ { , , } . Figure 6 illustrates the main cases to consider. Thefirst row displays the cases where g ( u ) = 1 , the second and third rows display the caseswhere g ( u ) = 2 , and the fourth and fifth rows display the cases where g ( u ) = 3 . u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u Figure 6: All cases for the proof of Observation 6.5. Solid edges are positive edges. Dashed edges arenegative edges.
Lemma 6.6. ( G, σ ) does not contain the configuration depicted in Figure 5(b).Proof. Suppose the contrary, i.e., assume that ( G, σ ) contains the configuration depictedin Figure 5(b). Because ( G, σ ) does not contain the configuration depicted in Figure 5(a)according to Lemma 6.4, note that the vertices v , v and v must be distinct. Let ( G ′ , σ ′ ) be the signed graph obtained from ( G, σ ) by deleting the vertices v , v and v and addingthe edge v v (if it was not already present). In case we do add this edge v v to ( G ′ , σ ′ ) ,then, as earlier, we choose its sign so that, in case G ′ = K , the signed graph ( G ′ , σ ′ ) is nota bad K . Then, by minimality, there is a homomorphism f : ( G ′ , σ ′ ) → ( SP +5 , (cid:3) + ) . Since ( SP +5 , (cid:3) + ) is transitive, we may assume that f ( v ) = ∞ . Moreover, since ( SP +5 −∞ , (cid:3) + ) = SP , (cid:3) ) is sp-edge-transitive and sp-isomorphic to ( SP , V ( SP ) \ (cid:3) ) , we may assumethat f ( v ) = 1 and f ( v ) = 2 . Finally, we may (if needed) switch some vertices of theconfiguration so that σ ( v v ) = σ ( v v ) = σ ( v v ) = + . By Observation 6.5, we can extend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) unless f ( v ) = 2 and σ ( v v ) = σ ( v v ) = σ ( v v ) . This leads us to the following two cases:1. f ( v ) = 2 and σ ( v v ) = σ ( v v ) = σ ( v v ) = + .In this case, we set f ( v ) = 2 in ( G ′ , σ ′ ) . The homomorphism can then be extendedto ( G, σ ) by setting f ( v ) = 3 and f ( v ) = ∞ .2. f ( v ) = 2 and σ ( v v ) = σ ( v v ) = σ ( v v ) = − .In this case, in ( G ′ , σ ′ ) we first switch v and v before setting f ( v ) = 3 . Thehomomorphism can then be extended to ( G, σ ) by setting f ( v ) = 5 and f ( v ) = ∞ .In all cases, it is thus possible to extend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) .This is a contradiction.In order to reduce the next configuration, we need the following: Observation 6.7.
For every two distinct i, j in V ( SP ) and { α, β } = { + , −} , we have N α ( i ) ∩ N β ( j ) = ∅ .Proof. Due to the structure of SP , it is sufficient to prove the statement for i = 1 and j ∈ { , } . In both cases we have N + (1) ∩ N − ( j ) = { } and N − (1) ∩ N + ( j ) = { j + 1 } Lemma 6.8. ( G, σ ) does not contain the configuration depicted in Figure 5(c).Proof. Suppose the contrary, i.e., assume that ( G, σ ) contains the configuration depicted inFigure 5(c). As ( G, σ ) does not contain the configuration depicted in Figure 5(b) accordingto Lemma 6.6, the vertices v and v must be distinct in ( G, σ ) , and similarly for the vertices v and v . Let ( G ′ , σ ′ ) be the signed graph obtained from ( G, σ ) by deleting the vertices v , v , v and v and adding the edge v v and v v (if they were not already present).As before, we choose the sign of each edge we add in such a way that ( G ′ , σ ′ ) is not a bad K . This way, by minimality, there is a homomorphism f : ( G ′ , σ ′ ) → ( SP +5 , (cid:3) + ) . Weknow that f ( v ) = f ( v ) and f ( v ) = f ( v ) . This brings us to two cases without loss ofgenerality.1. f ( v ) / ∈ { f ( v ) , f ( v ) } .Without loss of generality, we may assume f ( v ) = 1 and f ( v ) = ∞ . Assume that f ( v ) = j for some j
6∈ {∞ , } . Start by switching some of v , v , v , v (if needed)to make sure that σ ( v v ) = σ ( v v ) = σ ( v v ) = σ ( v v ) = + . Set f ( v ) = ∞ . Now, choose some i ∈ N σ ( v v ) ( f ( v )) \ {∞ , j } and set f ( v ) = i .According to Observation 6.5, there is an sp-homomorphism g of the signed pathinduced by the vertices v , v , v , v such that g ( v ) = j and g ( v ) = i . We can nowextend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) by setting f ( v ) = g ( v ) and f ( v ) = g ( v ) . 22. f ( v ) ∈ { f ( v ) , f ( v ) } . Up to the left/right symmetry, we may also assume that f ( v ) ∈ { f ( v ) , f ( v ) } , i.e. { f ( v ) , f ( v ) } = { f ( v ) , f ( v ) } .We here consider two subcases:(a) f ( v ) = f ( v ) and f ( v ) = f ( v ) .Without loss of generality, assume f ( v ) = f ( v ) = ∞ and f ( v ) = f ( v ) = 1 .Start by switching v , v (if needed) to make sure that σ ( v v ) = σ ( v v ) = + . • If the cycle v v v v v is positive, then switch v (if needed) to make surethat σ ( v v ) = σ ( v v ) and σ ( v v ) = σ ( v v ) . Now choose i ∈ N σ ( v v ) (1) \{∞} , j ∈ N σ ( v v ) (1) \ {∞ , i } and set f ( v ) = i and f ( v ) = j . Accordingto Observation 6.7, there is a way to extend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) by correctly setting f ( v ) , f ( v ) . • If the cycle v v v v v is negative, then switch some of v , v (if needed) tomake sure that σ ( v v ) = + and σ ( v v ) = − . Up to exchanging ( v , v ) with ( v , v ) , we may assume that σ ( v v ) = σ ( v v ) and σ ( v v ) = σ ( v v ) .On the one hand, if σ ( v v ) = σ ( v v ) = + , then set f ( v ) = 2 , f ( v ) = 3 and f ( v ) = 4 . On the other hand, if σ ( v v ) = σ ( v v ) = − , then set f ( v ) = 2 , f ( v ) = 5 and f ( v ) = 3 . Now Observation 6.7 tells us thatthere is a way to extend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) bycorrectly setting f ( v ) .(b) f ( v ) = f ( v ) and f ( v ) = f ( v ) .Without loss of generality, assume f ( v ) = f ( v ) = ∞ and f ( v ) = f ( v ) = 1 .Start by switching some of v , v , v , v (if needed) to make sure that σ ( v v ) = σ ( v v ) = σ ( v v ) = + and σ ( v v ) = − . Set f ( v ) = 2 and f ( v ) = 3 if σ ( v v ) = + , and f ( v ) = 4 otherwise. In both cases, according to Observation 6.5, there is a way to extend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) by correctly setting f ( v ) and f ( v ) .Thus, in all cases, it is possible to extend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) .This is a contradiction.We need two more results to deal with the last configuration in Figure 5. The firstone deals with two particular signed graphs, ( X, ϕ ) and ( X, ϕ ′ ) . Let ( X, ϕ ) be the signedgraph of order consisting of a -cycle uvwu and of a vertex x adjacent to w , where ϕ isa signature such that uvwu is a positive cycle. Let ( X, ϕ ′ ) be the signed graph obtainedfrom ( X, ϕ ) by switching the vertex w . Observation 6.9.
Let g be a partial sp-homomorphism from ( X, ϕ ) to ( SP +5 , (cid:3) + ) , whereonly u and v have an image under g . Then, up to switching the vertex w , it is possible toextend g to an sp-homomorphism from ( X, ϕ ) to ( SP +5 , (cid:3) + ) satisfying the following:(a) if { g ( u ) , g ( v ) } = {∞ , i } and ϕ ( wx ) = ϕ ( uw ) , then g ( x ) / ∈ { i − , i + 1 } ;(b) if { g ( u ) , g ( v ) } = {∞ , i } and ϕ ( wx ) = ϕ ( uw ) , then g ( x ) / ∈ {∞ , i } ;(c) if { g ( u ) , g ( v ) } = { i, i + 1 } and ϕ ( wx ) = ϕ ( uw ) , then g ( x ) = ∞ ; ∞ v w , x ∞ , , , u ∞ v w ∅ x ∅ u ∞ v w , x , , , u ∞ v w ∅ x ∅ u v w ∞ x , , , , u v w x , u v w ∞∅ u v w x ∞ , , u v w x ∞ , , u v w x , u v w x , u v w x ∞ , , Figure 7: All cases for the proof of Observation 6.9. Solid edges are positive edges. Dashed edges arenegative edges. (d) if { g ( u ) , g ( v ) } = { i, i + 1 } and ϕ ( wx ) = ϕ ( uw ) , then g ( x ) / ∈ { i, i + 1 , i + 3 } ;(e) if { g ( u ) , g ( v ) } = { i, i + 2 } and ϕ ( wx ) = ϕ ( uw ) , then g ( x ) / ∈ { i + 2 , i + 4 } ;(f ) if { g ( u ) , g ( v ) } = { i, i + 2 } and ϕ ( wx ) = ϕ ( uw ) , then g ( x ) / ∈ { i, i − } .Proof. Due to the symmetric structure of ( SP +5 , (cid:3) + ) , it is sufficient to consider the casesdepicted in Figure 7. For each considered value of { g ( u ) , g ( v ) } and ϕ ( wx ) , ϕ ( uw ) , we givetwo signatures on X ( ϕ and ϕ ′ ) and the corresponding possible values of g ( w ) and g ( x ) .The second result we need deals with two additional signed graphs, ( Y, ϕ ) and ( Y, ϕ ′ ) ,obtained from ( X, ϕ ) and ( X, ϕ ′ ) , respectively, by adding a new vertex y adjacent to x through a positive edge. Observation 6.10.
Let g be a partial sp-homomorphism of ( Y, ϕ ) to ( SP +5 , (cid:3) + ) , whereonly u , v and y have an image under g . Then, it is possible to extend g to an sp-homomorphism of ( Y, ϕ ) or ( Y, ϕ ′ ) to ( SP +5 , (cid:3) + ) .Proof. Observe that, for ∞ and any other vertex in ( SP +5 , (cid:3) + ) , the union of their positiveneighborhoods is V ( SP +5 ) . Moreover, note that, in ( SP +5 , (cid:3) + ) , we have ∀ i ∈ V ( SP ) , N + ( i ) ∪ N + ( i + 1) ∪ N + ( i + 2) = V ( SP +5 ) . The result now follows from Observation 6.9.24e are now ready to reduce the final configuration.
Lemma 6.11. ( G, σ ) does not contain the configuration depicted in Figure 5(d).Proof. Suppose the contrary, i.e., assume that ( G, σ ) contains the configuration depictedin Figure 5(d). Let ( G ′′ , σ ′′ ) be the signed graph obtained from ( G, σ ) by adding the edges v v , v v , v v and v v . Note that these edges were not already present in ( G, σ ) , since ( G, σ ) cannot have -cycles according to Lemma 6.6. Also let ( G ′ , σ ′ ) be the signed graphobtained from ( G ′′ , σ ′′ ) by deleting the vertices v , v , v , v , v and v . Observe thatif a connected component of G ′ is isomorphic to K , then G must have a cut-vertex, whichis not possible by Lemma 6.2. Thus, we can freely choose the signs of the edges we havejust added, without caring of whether a bad K is created. Precisely, we assign signs to v v , v v , v v , v v in such a way that the -cycle v v v v is negative, while the -cycles v v v v , v v v v and v v v v are positive.By minimality of ( G, σ ) , there is a homomorphism f : ( G ′ , σ ′ ) → ( SP +5 , (cid:3) + ) . Because ( SP +5 , (cid:3) + ) is edge-transitive, without loss of generality we may assume f ( v ) = 1 and f ( v ) = 3 . Note also that because the cycle v v v v is negative and v v is a negativeedge (due to the images of v , v in ( SP +5 , σ ) ), we can, if needed, switch v to ensure σ ( v v ) = σ ( v v ) = + . We can also switch v and/or v (if needed) to ensure σ ( v v ) = σ ( v v ) = + .Note that the signed subgraphs induced by { v , v , v , v } and { v , v , v , v } areexactly the signed graphs ( X, ϕ ) or ( X, ϕ ′ ) described in Observation 6.9. If one of themdoes not fall into Observation 6.9(d), then at most five values are forbidden at f ( v ) . Thusit is possible to extend f to { v , v , v } a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) .Otherwise, both of them satisfy the requirements of Observation 6.9(d), and we canalso extend f to { v , v , v } by setting (for example) f ( v ) = ∞ .Now, observe that { v , v , v , v , v } induces the graph Y . By Observation 6.10, wecan extend f to { v , v } regardless of the value of the values of f ( v ) , f ( v ) and f ( v ) .Finally, we extend f to a homomorphism of ( G, σ ) to ( SP +5 , (cid:3) + ) by assigning f ( v ) = ∞ if f ( v ) = ∞ and f ( v ) = 2 otherwise, which is a contradiction.The proof of Theorem 1.13 now follows from the fact that every subcubic graph differ-ent from K must have minimum degree or , or must contain one of the configurationsdepicted in Figure 5. Then, the previous lemmas imply that ( G, σ ) cannot exist, a contra-diction.
7. Conclusion
In this work, we have investigated the signed chromatic number of particular classesof graphs, namely planar graphs, triangle-free planar graphs, K n -minor-free graphs, andgraphs with bounded maximum degree. We have mainly considered general bounds (Theo-rems 1.10 and 1.13) for some of these classes, and the uniqueness of bounds (Theorems 1.8and 1.9) for the others. While some of our results are original ones, other ones extendknown results from the literature.Most of our results yield interesting research perspectives for the future, either becausethey are not tight yet, or because they lead to interesting side questions. In particular,we wonder how the bounds in Theorems 1.10 and 1.11 should be sharpened. RegardingTheorem 1.13, it would be interesting to determine whether ( SP +5 , (cid:3) + ) is the only bound oforder for subcubic graphs. Regarding Theorems 1.8 and 1.9, it would be, more generallyspeaking, of prime importance to understand better the signed chromatic number of planar25raphs, for which the currently best known lower and upper bounds are rather distant. Aninteresting more general question as well, could be to consider how the signed chromaticof a planar graph relates to its girth. That is, studying χ s ( P g ) for any g ≥ . Acknowledgement
The authors were partly supported by ANR project HOSIGRA (ANR-17-CE40-0022),by IFCAM project “Applications of graph homomorphisms” (MA/IFCAM/18/39) and bythe MUNI Award in Science and Humanities of the Grant Agency of Masaryk university.
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