Homomorphisms of signed planar graphs
aa r X i v : . [ c s . D M ] J a n Homomorphisms of signed planar graphs Pascal Ochem, Alexandre Pinlou LIRMM, Universit´e Montpellier 2, CNRS, France.
Sagnik Sen
LaBRI, Universit´e de Bordeaux, CNRS, France.
Abstract
Signed graphs are studied since the middle of the last century. Recently, the notionof homomorphism of signed graphs has been introduced since this notion captures anumber of well known conjectures which can be reformulated using the definitionsof signed homomorphism.In this paper, we introduce and study the properties of some target graphs forsigned homomorphism. Using these properties, we obtain upper bounds on thesigned chromatic numbers of graphs with bounded acyclic chromatic number andof signed planar graphs with given girth.
Key words:
Signed graphs, Homomorphisms, Discharging method.
The class of signed graphs is a natural graph class where the edges are eitherpositive or negative. They were first introduced to handle problems in socialpsychology: positive edges link friends whereas negative ones link enemies.In the area of graph theory, they have been used as a way of extending classicalresults in graph coloring such as Hadwiger’s conjecture. Guenin [3] introduced
Email addresses:
[email protected] (Pascal Ochem),
[email protected] (Alexandre Pinlou),
[email protected] (SagnikSen).
URLs: (Pascal Ochem), (Alexandre Pinlou). This work was partially supported by the ANR grant EGOS 12-JS02-002-01 andby the PEPS grant HOGRASI. Second affiliation: D´epartement de Math´ematiques et Informatique Appliqu´es,Universit´e Paul-Val´ery, Montpellier 3, France.a) (b)Fig. 1. Two equivalent signified graphs. the notion of signed homomorphism for its relation with a well known conjec-ture of Seymour. In 2012, this notion has been further developed by Naserasret al. [7] as this theory captures a number of well known conjectures which canbe reformulated using the definitions of signed homomorphism. In this paper,we study signed homomorphisms for themselves.A signified graph ( G, Σ) is a graph G with an assignment of positive (+1)and negative ( −
1) signs to its edges where Σ is the set of negative edges.In all the figures, negative edges are drawn with dashed edges. Figure 1(a)gives an example of signified graph.
Resigning a vertex v of a signified graph( G, Σ) corresponds to give the opposite sign to the edges incident to v . Givena signified graph ( G, Σ) and a set of vertices X ⊆ V ( G ), the graph obtainedfrom ( G, Σ) by resigning every vertex of X is denoted by ( G, Σ ( X ) ).Two signified graphs ( G, Σ ) and ( G, Σ ) are said to be equivalent if we canobtain ( G, Σ ) from ( G, Σ ) by resigning some vertices of ( G, Σ ), i.e., Σ =Σ ( X )1 for some X ⊆ V ( G ); in such a case, we use the notation ( G, Σ ) ∼ ( G, Σ ) (see Figure 1 for an example of equivalent signified graphs). Eachequivalence class defined by the resigning process is called a signed graph andcan be denoted by any member of its class. We might simply use ( G ) for asignified/signed graph when its set of negative edges is clear from the context,while G refers to its underlying unsigned graph.An m -edge-colored graph G is a graph where the vertices are linked by edges E ( G ) of m types. In other words, there is a partition E ( G ) = E ( G ) ∪ . . . ∪ E m ( G ) of the edges of G , where E j ( G ) contains all edges of type j . Sincesignified graphs are defined with two types of edges (i.e. positive and negativeedges), they correspond to 2-edge-colored graphs. In this paper, known andnew results on 2-edge-colored graphs are stated in terms of signified graphs.Given two graphs ( G, Σ) and ( H, Λ), ϕ is a signified homomorphism of ( G, Σ)to ( H, Λ) if ϕ : V ( G ) −→ V ( H ) is a mapping such that every edge of ( G, Σ) ismapped to an edge of the same sign of ( H, Λ). Given two graphs ( G, Σ ) and( H, Λ ), we say that there is a signed homomorphism ϕ of ( G, Σ ) to ( H, Λ ) ifthere exists ( G, Σ ) ∼ ( G, Σ ) and ( H, Λ ) ∼ ( H, Λ ) such that ϕ is a signifiedhomomorphism of ( G, Σ ) to ( H, Λ ). Lemma 1 If ( G, Σ) admits a signed homomorphism to ( H, Λ) , then thereexists ( G, Σ ′ ) ∼ ( G, Σ) such that ( G, Σ ′ ) admits a signified homomorphism to ( H, Λ) . roof. Since ( G, Σ) admits a signed homomorphism to ( H, Λ), this impliesthat there exist ( G, Σ ′′ ) ∼ ( G, Σ), ( H, Λ ′ ) ∼ ( H, Λ) and a signified homomor-phism ϕ of ( G, Σ ′′ ) to ( H, Λ ′ ). Let X ⊆ V ( H ) be the subset of vertices of H such that ( H, Λ) = ( H, Λ ′ ( X ) ). Now let Y = { v ∈ V ( G ) | ϕ ( v ) ∈ X } . Let( G, Σ ′ ) = ( G, Σ ′′ ( Y ) ); it is clear that ϕ is a signified homomorphism of ( G, Σ ′ )to ( H, Λ). ✷ As a consequence of the above lemma, when dealing with signed homomor-phisms, we will not need to resign the target graph.The signified chromatic number χ ( G, Σ) of the graph ( G, Σ) is the mini-mum order (number of vertices) of a graph ( H, Λ) such that ( G, Σ) admitsa signified homomorphism to ( H, Λ). Similarly, the signed chromatic number χ s ( G, Σ) of the graph ( G, Σ) is the minimum order of a graph ( H, Λ) such that( G, Σ) admits a signed homomorphism to ( H, Λ); equivalently, χ s ( G, Σ) =min { χ ( G, Σ ′ ) | ( G, Σ ′ ) ∼ ( G, Σ) } .The signified chromatic number χ ( G ) of a graph G is defined as χ ( G ) =max { χ ( G, Σ) | Σ ⊆ E ( G ) } . The signified chromatic number χ ( F ) of a graphclass F is defined as χ ( F ) = max { χ ( G ) | G ∈ F } . The signed chromaticnumbers of a graph and a graph class are defined similarly.Another equivalent definition of the signified chromatic numbers can be givenby defining the signified coloring. A signified coloring of a signified graph ( G )is a proper vertex-coloring ϕ of G such that if there exist two edges uv and xy with ϕ ( u ) = ϕ ( x ) and ϕ ( v ) = ϕ ( y ), then these two edges have the samesign. Hence, the signified chromatic number of the signified graph ( G ) is theminimum number of colors needed for a signified coloring of ( G ).In this paper, we studied signified and signed homomorphisms of outerplanarand planar graphs of given girth. The paper is organized as follows. We intro-duce the notation in Section 2. Section 3 is devoted to introduce and study theproperties of several families of target graphs, namely the Anti-twinned graph AT ( G, Σ), the signified Zielonka graph ZS k , the signified Paley graph SP q , andthe signified Tromp Paley graph T r ( SP q ). We study signified homomorphismsof planar graphs (resp. outerplanar graphs) in Section 4 and we provide lowerand upper bounds on the signified chromatic number. We get upper boundson the signed chromatic number of planar graphs (resp. outerplanar graphs)of given girth in Section 5. We finally conclude in Section 6. For a vertex v of a signified graph ( G ), d ( G ) ( v ) denotes the degree of v . Theset of positive neighbors of v is denoted by N +( G ) ( v ) and the set of negativeneighbors of v is denoted by N − ( G ) ( v ). Thus, the set of neighbors of v , denotedby N ( G ) ( v ), is N ( G ) ( v ) = N +( G ) ( v ) ∪ N − ( G ) ( v ). A vertex of degree k (resp. at least k , at most k ) is called a k -vertex (resp. ≥ k -vertex, ≤ k -vertex). If a vertex u isadjacent to a k -vertex ( ≥ k -vertex, ≤ k -vertex) v , then v is a k -neighbor (resp. ≥ k -neighbor, ≤ k -neighbor) of u . A path of length k (i.e. formed by k edges) is3 G , Σ ) u v u v ( G , Σ ) Fig. 2. The anti-twinned graph AT ( G, Σ). called a k -path. Given a planar graph G with its embedding in the plane anda vertex v of G , we say that a sequence ( u , u , · · · , u k ) of neighbors of v are consecutive if u , u , · · · , u k appear consecutively around v in G (clockwise orcounterclockwise). In this section, our goal is not only to find target graphs that will give therequired upper bounds of our results of Sections 4 and 5. We aim at describingseveral families of target graphs that may be useful for signified and signedhomomorphisms and we determine their properties. To this end, we describebelow the
Anti-twinned graph construction, the signified Zielonka graph ZS k ,the signified Paley graph and the Tromp signified Paley graph . Let ( G, Σ) be a signified graph and let ( G , Σ ) and ( G , Σ ) be two isomorphiccopies of ( G, Σ). In the following, given a vertex u ∈ V ( G ), we denote u i thecorresponding vertex of u in the isomorphic copy ( G i , Σ i ) of ( G, Σ). We definethe anti-twinned graph AT ( G, Σ) = ( H, Λ) on 2 | V ( G ) | vertices as follows: • V ( H ) = V ( G ) ∪ V ( G ) • E ( H ) = E ( G ) ∪ E ( G ) ∪ { u i v − i : uv ∈ E ( G ) }• Λ = Σ ∪ Σ ∪ { u i v − i : uv ∈ E ( G ) \ Σ } Figure 2 illustrates the construction of AT ( G, Σ). We can observe that forevery u i ∈ V ( G i ), there is no edge between u i and u − i . By construction wehave the following property: ∀ u i ∈ AT ( G, Σ) : N + ( u i ) = N − ( u − i ) and N − ( u i ) = N + ( u − i )Such pairs of vertices are called anti-twin vertices , and for any u ∈ AT ( G, Σ)we denote by atw( u ) the anti-twin vertex of u . Remark that atw(atw( u )) = u .This notion can be extended to sets in a standard way: for a given W ⊆ V ( G i ), W = { v , v , . . . , v k } , then atw( W ) = { atw( v ) , atw( v ) , . . . , atw( v k ) } .We say that a signified graph is anti-twinned if it is the anti-twinned graph ofsome signified graph. Observation 2
A signified graph is anti-twinned if and only if each of itsvertices has a unique anti-twin. emma 3 A graph ( G, Σ) admits a signed homomorphism to ( H, Λ) if andonly if ( G, Σ) admits a signified homomorphism to AT ( H, Λ) . Proof.
Let ϕ be a signed homomorphism of ( G, Σ) to ( H, Λ). This implies that ϕ is a signified homomorphism of ( G, Σ ) to ( H, Λ), where ( G, Σ ) ∼ ( G, Σ).Let X ⊆ V ( G ) be the subset of vertices of G such that ( G, Σ) = ( G, Σ ( X )1 ).By definition of the resigning process, only the edges of the edge-cut between V ( G ) \ X and X get their sign changed.Let ϕ ′ : V ( G ) → V ( AT ( H )) be defined as follows: ϕ ′ ( u ) = atw( ϕ ( u )) , if u ∈ X,ϕ ( u ) , otherwise . By construction of AT ( H, Λ), if u and v induce an edge of sign s , then u andatw( v ) induce an edge of sign − s . Therefore, it is easy to see that ϕ ′ is asignified homomorphism of ( G, Σ) to AT ( H, Λ). This proves the only if part.For the if part, suppose that ( G, Σ) admits a signified homomorphism ψ to AT ( H, Λ). The signified graph AT ( H, Λ) is obtained from two isomorphiccopies ( H , Λ ) and ( H , Λ ) of ( H, Λ). Let Y ⊆ V ( G ) be the subset of verticesof G such that, for all y ∈ Y , ψ ( y ) is a vertex of H . Let ( G, Σ ) = ( G, Σ ( Y ) ).Let ψ ′ : V ( G ) → V ( AT ( H )) be defined as follows: ψ ′ ( u ) = atw( ψ ( u )) , if u ∈ Y ,ψ ( u ) , otherwise . It is easy to see that ψ ′ is a signified homomorphism of ( G, Σ ) to AT ( H, Λ)such that every vertex maps to a vertex of H . Then ψ ′ is a signified homomor-phism from ( G, Σ ) to ( H, Λ) and thus ( G, Σ) admits a signed homomorphismto ( H, Λ). ✷ Corollary 4 If ( G, Σ) admits a signified homomorphism to an anti-twinnedgraph T , then we have:(1) χ s ( G ) ≤ | V ( T ) | .(2) ( G, Σ ′ ) admits a signified homomorphism to T for every ( G, Σ ′ ) ∼ ( G, Σ) .3.2 The signified Zielonka graph ZS k The Zielonka graph Z k was introduced by Zielonka [12] in the theory ofbounded timestamp systems. Alon and Marshall [1] adapted this construc-tion to signified graphs to obtain the signified Zielonka graph ZS k . They usedthis graph to get bounds on the signified chromatic number of graphs thatadmits an acyclic k -coloring.Let us describe the construction of the signified Zielonka graph ZS k . Everyvertex is of the form ( i ; α , α , . . . , α k ) where 1 ≤ i ≤ k , α j ∈ { +1 , − } for j = i and α i = 0. There are clearly k · k − vertices in this graph. For i = j ,5 Fig. 3. The signified graph SP . there is an edge between the vertices ( i ; α , α , . . . , α k ) and ( j ; β , β , . . . , β k )and the sign of this edge is given by the product α j × β i . Proposition 5
The graph ZS k is anti-twinned. Proof.
By Observation 2, we have to show that every vertex has an anti-twin.We claim that the anti-twin of the vertex v = ( i ; α , α , . . . , α k ) is the vertex v ′ = ( i ; − α , − α , . . . , − α k ). Indeed, v and v ′ are not adjacent and it is easyto check that for every edge uv , the edge uv ′ exists, and that uv and uv ′ haveopposite signs. ✷ SP q In the remaining, q is any prime power such that q ≡ F q of order q . Let g be a generator ofthe field F ∗ q . For every v ∈ F ∗ q , let sq : F ∗ q → {− , +1 } be the function square defined as sq( v ) = +1 if v is a square and sq( v ) = − v is a non-square.Note that sq( g t ) = ( − t since g is necessarily a non-square.The Paley graph P q is the undirected graph with vertex set V ( P q ) = F q andedge set E ( P q ) = { xy | sq( y − x ) = +1 } . Since − F q , sq( x − y ) =sq( y − x ) and therefore the definition of an edge is consistent. We also knowthat a Paley graph is self-complementary [10] and edge-transitive.A k -regular graph G with n vertices is said to be strongly regular if (1) ev-ery two adjacent vertices have λ common neighbors and (2) every two non-adjacent vertices have µ common neighbors. Such a graph is said to be astrongly regular graph with parameters ( n, k, λ, µ ). Paley graphs P q are knownto be strongly regular graphs with parameters ( q, q − , q − , q − ).For any prime power q ≡ signified Paley graph SP q =( K q , Σ) as the complete graph on q vertices with V ( SP q ) = F q and Σ = { xy | sq( y − x ) = − } . That is, SP q is obtained from the Paley graph P q byreplacing the non-edges by negative edges. Figure 3 represents the signifiedPaley graph SP . Since P q is edge-transitive and self-complementary, SP q isclearly edge-transitive. 6 ∞ G ∞ u v G v Fig. 4. The signified Tromp graph
T r ( G, Σ).
T r ( SP q )Given an oriented graph −→ G , Tromp [11] proposed a construction of an orientedgraph T r ( G ) called Tromp graph. We adapt this construction to signifiedgraphs as follows.For a given graph ( G, Σ), let us denote ( G + , Σ) the graph obtained from ( G, Σ)by adding a universal vertex positively linked to all the vertices of ( G, Σ).Then, the
Tromp signified graph
T r ( G, Σ) of ( G, Σ) is defined to be the anti-twinned graph of ( G + , Σ), that is
T r ( G, Σ) ∼ = AT ( G + , Σ). Figure 4 illustratesthe construction of
T r ( G, Σ).By construction,
T r ( G, Σ) is obtained from two isomorphic copies ( G , Σ )and ( G , Σ ) of ( G, Σ) plus 2 vertices ∞ and ∞ .In the remainder, we focus on the specific graph family obtained by applyingthe Tromp’s construction to the signified Paley graph SP q .We consider the Tromp signified Paley graph T r ( SP q ) on 2 q + 2 vertices ob-tained from SP q . In the remainder of this paper, the vertex set of T r ( SP q ) is V ( T r ( SP q )) = { , , . . . , q − , ∞ , , , . . . , q − , ∞ } where { i , i , . . . ,q − i } is the vertex set of the isomorphic copy SP iq of SP q ( i ∈ { , } ); thus, forevery u i ∈ { i , i , . . . , q − i , ∞ i } , we have atw( u i ) = u − i . In addition, for every u ∈ V ( T r ( SP q )), we have by construction | N + T r ( SP q ) ( u ) | = | N − T r ( SP q ) ( u ) | = q .Let i, j ∈ { , } and u, v ∈ F q . If i = j , then u i and v j are in the sameisomorphic copy of SP q in T r ( SP q ); in this case, the sign of the edge u i v j issq( u − v ) by definition of SP q . If i = j , then u i and v j are in distinct isomorphiccopies of SP q in T r ( SP q ); in this case, the sign of the edge u i v j is − sq( u − v ).Therefore, in both cases, the sign of the edge u i v j issq( u − v ) × ( − i + j . (1)Let i, j ∈ { , } and v ∈ F q . If i = j , then the sign of the edge ∞ i v j is +1,while it is − i = j . Therefore, in both case, the sign of the edge ∞ i v j is 7 − i + j . (2)The graph T r ( SP q ) has remarkable symmetry and some useful properties givenbelow. Lemma 6
The signified graph
T r ( SP q ) is vertex-transitive. Proof.
The mapping γ : V ( T r ( SP q )) → V ( T r ( SP q )) defined as γ ( u i ) = u − i is clearly an automorphism of T r ( SP q ).Recall first that SP q is edge-transitive and so vertex-transitive. If ϕ is anautomorphism of SP q , we can define the corresponding automorphism γ of T r ( SP q ) as: γ : u i → u i if u = ∞ ( ϕ ( u )) i if u = ∞ We eventually define the mapping γ : V ( T r ( SP q )) → V ( T r ( SP q )) as: γ : u i → ∞ i if u = 00 i if u = ∞ ( u − ) i if u is a non-zero square( u − ) − i if u is a non squareLet S be the sign of the edge u i v j . To prove that γ is an automorphism of T r ( SP q ), we will show that γ maps u i v j to an edge of sign S ′ = S .Let g be a generator of the field F q . Any vertex v of SP q is an element of F q and therefore v = g t for some t when v = 0. Recall that sq( g t ) = ( − t . • When u, v ∈ V ( T r ( SP q )) are neither 0 nor ∞ , we have u i v j = ( g t ) i ( g t ′ ) j for some t and t ′ . If t is even, that is g t is a square, then γ (( g t ) i ) = ( g − t ) i ;otherwise, t is odd and thus γ (( g t ) i ) = ( g − t ) − i . Let i ′ = i + t (mod 2)and j ′ = j + t ′ (mod 2). Clearly, if t is even (resp. t ′ ) is even, then i ′ = i (resp. j ′ = j ); otherwise we have i ′ = 1 − i (resp. j ′ = 1 − j ). Therefore, thefunction γ maps the edge ( g t ) i ( g t ′ ) j to the edge ( g − t ) i ′ ( g − t ′ ) j ′ . Now, let uscheck that the sign of these two edges is the same.By Equation (1), the sign of the edge u i v j is S = sq( g t − g t ′ ) × ( − i + j and the sign of the edge ( g − t ) i ′ ( g − t ′ ) j ′ is S ′ = sq( g − t − g − t ′ ) × ( − i ′ + j ′ . Wethen have: 8 = sq( g t − g t ′ ) × ( − i + j = sq( g t ′ − g t ) × ( − i + j = sq(( g − t − g − t ′ ) × g t + t ′ ) × ( − i + j = sq( g − t − g − t ′ ) × sq( g t + t ′ ) × ( − i + j = sq( g − t − g − t ′ ) × ( − t + t ′ × ( − i + j = sq( g − t − g − t ′ ) × ( − i + t + j + t ′ = sq( g − t − g − t ′ ) × ( − i ′ + j ′ = S ′ • If u = 0 and v = ∞ , it is clear that the edge u i v j maps to an edge of thesame sign. • Consider now the case u = 0 and v
6∈ { , ∞} . Let v = g t for some t . Thefunction γ maps the edge 0 i ( g t ) j on ∞ i ( g − t ) j ′ where j ′ = j + t (mod 2).By Equations 1 and 2, the sign of the edge 0 i ( g t ) j is S = sq( g t ) × ( − i + j and the sign of the edge ∞ i ( g − t ) j ′ is S ′ = ( − i + j ′ . We then have: S = sq( g t ) × ( − i + j = ( − t × ( − i + j = ( − i + j + t = ( − i + j ′ = S ′ • The case u = ∞ and v
6∈ { , ∞} is similar to the previous case.Therefore, the function γ maps any edge to an edge of the same sign.Combining the automorphisms γ , γ and γ easily proves that T r ( SP q ) isvertex-transitive. ✷ We define an anti-automorphism of a signified graph ( G, Σ) as a permutation ρ of the vertex set V ( G ) such that uv is a positive (resp. negative) edge if andonly if ρ ( u ) ρ ( v ) is a negative (resp. positive) edge. Lemma 7
The graph
T r ( SP q ) admits an anti-automorphism. Proof.
Let n be any non-square of F q . We define the mapping γ n : V ( T r ( SP q )) → V ( T r ( SP q )) as: γ n : u i → u i if u = ∞ ( n × u ) − i if u = ∞ Let us check that γ n maps every edge u i v j ∈ E ( T r ( SP q )) to an edge of oppositesign.Let u, v = ∞ . By definition, γ n maps u i v j to ( n × u ) − i ( n × v ) − j . By Equa-tions 1 and 2, the sign of the edge u i v j is S = sq( v − u ) × ( − i + j and the sign9f the edge ( n × u ) − i ( n × v ) − j is S ′ = sq( n × ( v − u )) × ( − − i +1 − j . S ′ = sq( n × ( v − u )) × ( − − i +1 − j = sq( n ) × sq( v − u ) × ( − i + j = − sq( v − u ) × ( − i + j = − S Now, let u = ∞ and v = ∞ . The mapping γ n maps ∞ i v j to ∞ i ( n × v ) − j . ByEquation 2, the sign of the edge ∞ i v j is S = ( − i + j and the sign of the edge ∞ i ( n × v ) − j is S ′ = ( − i +1 − j = − S . ✷ Lemma 8
If there exists an isomorphism ψ that maps the triangle ( u i , v j , w k ) to the triangle ( u ′ i ′ , v ′ j ′ , w ′ k ′ ) , then ψ can be extended to an automorphism of T r ( SP q ) . Proof.
There exists four types of triangles depending on the sign of theiredges. • Let us first consider the triangles ( u i , v j , w k ) and ( u ′ i ′ , v ′ j ′ , w ′ k ′ ) with 3 positiveedges. To prove that ψ can be extended to an automorphism of T r ( SP q ), itsuffices to prove that for every triangle ( u i , v j , w k ), there exists an automor-phism ψ ′ that maps ( u i , v j , w k ) to (0 , , ∞ ). Using the vertex transitivityof T r ( SP q ) (Lemma 6), there exists an automorphism ϕ that maps w k to ∞ . Then, since all positive edges incident to ∞ have their extremities in SP q , ϕ necessarily maps the edge u i v j to an edge u ′ v ′ in SP q . Since SP q is edge-transitive, we can finally map u ′ v ′ to 0 . • Consider now the triangles ( u j , v j , w k ) and ( u ′ j ′ , v ′ j ′ , w ′ k ′ ) with 3 negativeedges. Let T r ( SP q ) be the signified graph obtained from T r ( SP q ) by chang-ing the sign of every edge. By Lemma 7, T r ( SP q ) is isomorphic to T r ( SP q ).By the previous item, there exists an automorphism that maps ( u j , v j , w k )to ( u ′ j ′ , v ′ j ′ , w ′ k ′ ) in T r ( SP q ), and thus in T r ( SP q ). • Finally consider the triangles ( u i , v j , w k ) and ( u ′ i ′ , v ′ j ′ , w ′ k ′ ) with one edge ofsign S and 2 edges of sign − S . Let u i and u ′ i ′ be the vertex incident to theedges of sign − S . Consider the triangles (atw( u i ) , v j , w k ) and (atw( u ′ i ′ ) , v ′ j ′ ,w ′ k ′ ); they have 3 edges of sign S . By the two previous cases, there exists anautomorphism ψ that maps (atw( u j ) , v j , w k ) to (atw( u ′ j ′ ) , v ′ j ′ , w ′ k ′ ). Since ψ preserves anti-twinning, ψ also maps u i to u ′ i ′ . ✷ A signed vector of size k is a k -tuple α = ( α , α , . . . , α k ) ∈ { +1 , − } k . For agiven signed vector α , its conjugate is the k -tuple ¯ α = ( − α , − α , . . . , − α k ).Given a sequence of k distinct vertices X k = ( v , v , . . . , v k ) of a signified graph( G, Σ) that induces a clique, a vertex u ∈ V ( G ) is an α -successor of X k if, forevery i ∈ { , , . . . , k } , the sign of the edge uv i is α i . The set of α -successorsof X k is denoted by S α ( X k ). 10onsider the signified graph SP depicted in Figure 3. For example, given α = (+1 , −
1) and X = (0 , α -successor of X , the vertex2 is an ¯ α -successor of X , we have S α ( X ) = { } and S ¯ α ( X ) = { } .A signified graph ( G ) has property P k,l if | S α ( X k ) | ≥ l for any sequence X k of k distinct vertices inducing a clique of G and for any signed-vector α of size k . Lemma 9 If SP q has property P n − ,k , then T r ( SP q ) has property P n,k . Proof.
Suppose that SP q has property P n − ,k and let α = ( α , α , . . . , α n )be a given signed vector. Let X = ( u , u , . . . , u n − , w ) be n distinct verticesinducing a clique of T r ( SP q ). We have to prove that X admits k α -successors.By noticing that S ¯ α ( X ) = atw( S α ( X )), we restrict the proof to the case α n = +1. We define X ′ = ( v , v , . . . , v n − , w ) such that v i = u i if u i w is apositive edge and v i = atw( u i ) if u i w is a negative edge. Hence, X ′ is a set of n distinct vertices of T r ( SP q ) such that S i v i ⊆ N + ( w ). By Lemma 6, T r ( SP q )is vertex-transitive and thus N + ( w ) ∼ = K q ∼ = SP q . Therefore the ( n − X ′′ = X ′ \ { w } = ( v , v , . . . , v n − ) form a subset of some V ( SP q ).Then by Property P n − ,k of SP q , there exist k ( α ′ , α ′ , . . . , α ′ n − )-successors x , x , . . . , x k of X ′′ in SP q , with α ′ i = α i (resp. α ′ i = − α i ) if v i = u i (resp. if v i = atw( u i )). The x i ’s are clearly positive neighbors of w and hence, they are( α ′ , α ′ , . . . , α ′ n − , α n )-successors of X ′ . So X has k α -successors. ✷ Lemma 10 If ( G, Σ) is a signified graph and T r ( G, Σ) has property P n,k , then AT ( G, Σ) has property P n,k − . Proof.
Recall that
T r ( G, Σ) is built from two isomorphic copies of ( G, Σ)plus two vertices ∞ and ∞ . The graph AT ( G, Σ) is obtained from
T r ( G, Σ)by removing both ∞ and ∞ . Now suppose T r ( G, Σ) has property P n,k . Thismeans that any n distinct vertices inducing a clique in T r ( G, Σ) has k α -successors for any signed n -vector α . Let X ⊆ V ( T r ( G, Σ)) be any sequenceof n distinct vertices inducing a clique in T r ( SP q ) such that both ∞ and ∞ do not belong to X . Then, for any signed n -vector α , the set of k α -successors S α ( X ) cannot contains both ∞ and ∞ . Then, it is clear that X has at least k − α -successors in AT ( G, Σ). ✷ Lemma 11 (1) SP q has properties P , q − and P , q − .(2) T r ( SP q ) has properties P ,q , P , q − , and P , q − .(3) AT ( SP q ) has properties P ,q − , P , q − , and P , max ( , q − ) . Proof. (1) These properties follow from the fact that the signified Paley graph SP q isbuilt from the Paley graph P q which is self-complementary, edge transitiveand strongly regular with parameters ( q, q − , q − , q − ).(2) T r ( SP q ) has property P ,q since it is vertex transitive by Lemma 6 and thevertex ∞ has q positive and q negative neighbors. The other propertiesfollow from (1) and Lemma 9. 113) These properties follow from (2) and Lemma 10. ✷ This section is devoted to study signified homomorphisms of planar graphsand outerplanar graphs.An acyclic k -coloring is a proper vertex-coloring such that each cycle has atleast three colors. In other words, the graph induced by any two color classesis a forest.In 1998, Alon and Marshall [1] proved the following (the signified graph ZS k has k · k − vertices and has been considered in Section 3.2): Theorem 12 ([1])
Let ( G, Σ) be such that G admits an acyclic k -coloring.Then ( G, Σ) admits a signified homomorphism to ZS k and thus χ ( G ) ≤ k · k − . Note that Theorem 12 is actually tight as shown by Huemer et al. [4] in 2008.The girth of a graph is the length of a shortest cycle. We denote by P g (resp. O g ) the class of planar graphs (resp. outerplanar graphs) with girth at least g (note that P is simply the class of planar graphs).Borodin [2] proved that every planar graph admits an acyclic 5-coloring. Wethus get the following from Theorem 12: Corollary 13
Every planar graph admits a signified homomorphism to ZS .We thus have χ ( P ) ≤ . In this same context, Montejano et al. [6] obtained in 2010 the following results:
Theorem 14 ([6]) (1) Every planar graph of girth at least admits a signified homomorphismto T r ( SP ) . We thus have χ ( P ) ≤ .(2) Every planar graph of girth at least admits a signified homomorphismto T r ( SP ) . We thus have χ ( P ) ≤ . In this section, we get the following new results. We obtain a first result onouterplanar graph with girth at least 4 (see Theorem 15) and a second one onplanar graph with girth at least 4 (see Theorem 16). The latter result givesa new upper bound on the signified chromatic number. We then construct aplanar graph with signified chromatic number 20 (see Theorem 18). We finallygive properties that must verify the target graphs for outerplanar and planargraphs.
Theorem 15
Every outerplanar graph with girth at least admits a signifiedhomomorphism to AT ( K ∗ ) , where K ∗ denotes the complete graph on verticeswith exactly one negative edge. Proof.
Assume by contradiction that there exists a counterexample to theresult and let ( H, Λ) be a minimal counterexample in term of number of ver-12ices.Suppose ( H ) contains a vertex u of degree at most 1. By minimality of ( H ),the graph ( H ′ ) = ( H \ { u } ) admits a signified homomorphism to AT ( K ∗ ).Since every vertex of AT ( K ∗ ) is incident to a positive and a negative edge, wecan extend the signified homomorphism to ( H ), a contradiction.Suppose that ( H ) contains two adjacent vertices u and v of degree 2. By mini-mality of ( H ), the graph ( H ′ ) = ( H \{ u, v } ) admits a signified homomorphismto AT ( K ∗ ). One can check that for every pair of (non necessarily distinct) ver-tices x and y of AT ( K ∗ ), there exist the 8 possible signified 3-paths. We cantherefore extend the signified homomorphism to ( H ), a contradiction.Pinlou and Sopena [9] showed that every outerplanar graph with girth at least k and minimum degree at least 2 contains a face of length l ≥ k with at least( l −
2) consecutive vertices of degree 2. Therefore, the counterexample ( H ) isnot an outerplanar graph of girth 4, a contradiction, that completes the proof. ✷ Theorem 16
Every planar graph with girth at least admits a signified ho-momorphism to AT ( SP ) . We thus have χ ( P ) ≤ . Let n ( G ) be the number of ≥ G . Let us define thepartial order (cid:22) . Given two graphs G and G , we have G ≺ G if and only ifone of the following conditions holds: • n ( G ) < n ( G ). • n ( G ) = n ( G ) and | V ( G ) | + | E ( G ) | < | V ( G ) | + | E ( G ) | .Note that the partial order (cid:22) is well-defined and is a partial linear extensionof the minor poset.Let ( H ) be a signified graph that does not admit a homomorphism to thesignified graph AT ( SP ) and such that its underlying graph H is a triangle-free planar graph which is minimal with respect to (cid:22) . In the following, H isgiven with its embedding in the plane. A weak 7-vertex u in H is a 7-vertexadjacent to four 2-vertices v , · · · , v and three ≥ w , w , w such that v , w , v , w , v , w , and v are consecutive. Lemma 17
The graph H does not contain the following configurations:(C1) a ≤ -vertex;(C2) a k -vertex adjacent to k -vertices for ≤ k ≤ ;(C3) a k -vertex adjacent to ( k −
1) 2 -vertices for ≤ k ≤ ;(C4) a k -vertex adjacent to ( k −
2) 2 -vertices for ≤ k ≤ ;(C5) a -vertex;(C6) a k -vertex adjacent to ( k −
3) 2 -vertices for ≤ k ≤ ;(C7) two vertices u and v linked by two distinct -paths, both paths having a -vertex as internal vertex;(C8) a -face wxyz such that x is 2-vertex, w and y are weak -vertices, and z is a k -vertex adjacent to ( k − ≤ k ≤ ; ≤ k ≤ v ′ v ′ k v v k v (a) C2 v ′ k v v k v v v ′ ≤ k ≤ (b) C3 ≤ k ≤ v ′ k v v v k v v v ′ (c) C4Fig. 5. Configurations C2–C4. Proof.
The drawing conventions for a configuration C k contained in a signedgraph ( H ) are the following. First note that, in Figures 5 and 6, we only drawthe underlying graph H of ( H ), i.e. we do not distinguish positive and negativeedges. The neighbors of a white vertex in H are exactly its neighbors in C k ,whereas a black vertex may have other neighbors in H . Two or more blackvertices in C k may coincide in a single vertex in H , provided they do not sharea common white neighbor. Configurations C2 - C8 are depicted in Figures 5and 6.For each configuration, we suppose that H contains the configuration and weconsider a signified triangle-free graph ( H ′ ) such that H ′ ≺ H . We only arguethat H ′ ≺ H for configuration C5. For every other configuration, we have that H ′ is a minor of H and thus H ′ ≺ H . Therefore, by minimality of ( H ), ( H ′ )admits a signified homomorphism f to AT ( SP ). Then we modify and extend f to obtain a signified homomorphism of ( H ) to AT ( SP ), contradicting thefact that ( H ) is a counterexample.By Lemma 11, AT ( SP ) satisfies properties P , , P , , and P , . Proof of configuration C1:
Trivial.
Proof of configuration C2:
Suppose that ( H ) contains the configurationdepicted in Figure 5(a) and f is a signified homomorphism of ( H ′ ) = ( H ) \{ v, v , · · · , v k } to AT ( SP ). For every i , if the edges vv i and v i v ′ i have the samesign (resp. different signs), then v must get a color distinct from atw( f ( v ′ i ))(resp. f ( v ′ i )). So, each v ′ i forbids at most one color for v . Thus there remainsan available color for v . Then we extend f to the vertices v i using property P , . Proof of configuration C3:
Suppose that ( H ) contains the configurationdepicted in Figure 5(b) and f is a signified homomorphism of ( H ′ ) = ( H ) \{ v, v , · · · , v k } to AT ( SP ). As shown in the proof of Configuration C2, each v ′ i forbids at most one color for v . So, we have at most 23 forbidden colors for v and by property P , , there remains at least one available color for v . Thenwe extend f to the vertices v i (2 ≤ i ≤ k ) using property P , . Proof of configuration C4:
Suppose that ( H ) contains the configurationdepicted in Figure 5(c) and f is a signified homomorphism of ( H ′ ) = ( H ) \{ v , · · · , v k } to AT ( SP ). As shown in the proof of Configuration C2, each v ′ i forbids at most one color for v . So, we have at most 10 forbidden colorsfor v and by property P , , there remains at least one available color in orderto recolor v . Then we extend f to the vertices v i (3 ≤ i ≤ k ) using property14 v v v (a) C5 v v v v v v ′ ≤ k ≤ v k v ′ k (b) C6 v wu v (c) C7 w u u u bd ac c w w c v v k − a a b b ≤ k ≤ (d) C8Fig. 6. Configurations C5–C8. P , . Proof of configuration C5:
Suppose that ( H ) contains the configurationdepicted in Figure 6(a). Let ( H ′ ) be the graph obtained from ( H ) by deletingthe vertex v and by adding, for every 1 ≤ i < j ≤
3, a new vertex v ij andthe edges v i v ij and v ij v j . Each of the 6 edges v i v ij gets the sign α i of the edge v i v in ( H ). As configuration C4 is forbidden in H , we have d H ( v i ) ≥ i ∈ { , , } . We have H ′ ≺ H since n ( H ′ ) < n ( H ). Clearly, H ′ is trianglefree. Hence, there exists a signified homomorphism f of ( H ′ ) to AT ( SP ).By P , , we can find an α -successor u of ( f ( v ) , f ( v ) , f ( v )) in AT ( SP )with α = ( α , α , α ). Now fix f ( v ) = u . Note that f restricted to V ( H ) is ahomomorphism of ( H ) to AT ( SP ). Proof of configuration C6:
Suppose that ( H ) contains the configurationdepicted in Figure 6(b) and f is a signified homomorphism of ( H ′ ) = ( H ) \{ v , . . . , v k } to AT ( SP ). As shown in the proof of Configuration C2, each v ′ i forbids at most one color for v . So, we have at most 3 forbidden colors for v and by property P , , there remains at least one available color for v . Then weextend f to the vertices v i (4 ≤ i ≤ k ) using property P , . Proof of configuration C7:
Suppose that ( H ) contains the configurationdepicted in Figure 6(c).If u and w have no common neighbor other than v and v , then we considerthe graph ( H ′ ) obtained from ( H ) \ { v , v } by adding the edge uw .If u and w have at least one other common neighbor v , then consider thegraph ( H ′ ) obtained from ( H ) \ { v , v } by adding a vertex v adjacent to u and w such that uv is negative and the sign of vw is the product of the signsof uv and v w . Therefore, we have at least two 2-paths linking u and w , onewhose both edges have the same sign and one whose edges have different sign.In both cases, H ′ is triangle free and is a minor of H , so that ( H ′ ) admits asignified homomorphism f to AT ( SP ). Also, in both cases, f ( u ) and f ( w )form an edge in AT ( SP ) since f ( u ) = f ( v ) and f ( u ) = f (atw( v )). Thus, thecoloring of ( H ) \ { v , v } induced by f can be extended to ( H ) using property P , . 150 , , ) 3 , , (1 + 2 √ , (1 + 3 √ (0 , , ) 2 , , (3 + √ , (3 + 4 √ (0 , , ) 2 , , (2 √ , (3 √ (0 , , (3 + 2 √ ) (3 + √ , (3 √ , (1 + 3 √ , (3 + 4 √ (0 , , (3 + 3 √ ) (3 + √ , (2 √ , (1 + 2 √ , (3 + 4 √ Tab. 7. Sets of the form (0 , , x ) having exactly 4 (+1 , +1 , +1)-successors in AT ( SP ). Proof of configuration C8:
Suppose that ( H ) contains the configurationdepicted in Figure 6(d). By Corollary 4(2), ( H ) admits a signified homomor-phism to AT ( SP ) if and only if every equivalent signature of ( H ) admits asignified homomorphism to AT ( SP ). So, by resigning a subset of vertices in { a, b, c, c , c } , we can assume that the edges da , ab , bc , cc , and cc are posi-tive. Consider a signified homomorphism f of ( H ′ ) = ( H \ { d } ) to AT ( SP ).The edge dc in ( H ) has to be negative, since otherwise f would be extendableto ( H ) by setting f ( d ) = f ( b ). Also, we must have f ( c ) = f ( a ), since otherwisewe could color d using property P , . Now, we show in the remaining of theproof that we can modify f such that f ( c ) = f ( a ). Let us define the signedvector α = (+1 , +1 , +1). We assume that f ( b ), f ( c ) and f ( c ) are distinct,since the case when they are not distinct is easier to handle.For 1 ≤ i ≤
3, let k i denote the color that is forbidden for c by w i , that is, k i = f ( w i ) if the edges of the 2-path linking c and w i have distinct signs and k i =atw( f ( w i )) otherwise. By property P , , the sequence X = ( f ( c ) , f ( c ) , f ( b ))has at least 4 α -successors. Assume that X has at least 5 α -successors. Then wecan give to c a color distinct from k , k , k , and f ( a ) that leads to f ( c ) = f ( a ).Assume now that X has exactly 4 α -successors. The graph AT ( SP ) containstwo copies of SP , namely SP and SP . Suppose that X is not containedin one copy of SP . We consider the graph T r ( SP ) obtained by adding theanti-twin vertices ∞ and ∞ to AT ( SP ). By Lemma 11, T r ( SP ) satisfies P , , so X admits at least 5 α -successors in T r ( SP ). Since X is not containedin one copy of SP of the subgraph AT ( SP ), the extra vertices ∞ and ∞ are not α -successors of X . This means that X has at least 5 α -successors in AT ( SP ), contradicting the hypothesis. So, without loss of generality, X isnecessarily contained in SP .We represent the field F by the numbers a + b √
2, where a and b are integersmodulo 5. Without loss of generality, we can assume that f ( c ) = 0 and f ( c ) = 1 since SP is edge-transitive. Table 7 gives the sequences X havingexactly 4 α -successors together with their 4 α -successors.We are now ready to modify f . We decolor the vertices a , b , and c . By property16 , , there exist at least two colors β and β ′ for b that are distinct from thecolors forbidden by the k vertices v , · · · , v k − , a , a , c , c . By previous discus-sions, the sequences (0 , , β ) and (0 , , β ′ ) must have exactly 4 α -successors, sowe can assume that { β, β ′ } ⊂ n , , , √ , √ o . Let us set f ( b ) = β .By property P , , we can color a such that f ( a ) is distinct from the colorsforbidden by u , u , u . Now, f is not extendable to c and d only if the 4 α -successors of (0 , , β ) are k , k , k , and f ( a ). In particular, k , k , and k have to be α -successors of (0 , , β ). Similarly, we can set f ( b ) = β ′ and ob-tain that k , k , and k have to be α -successors of (0 , , β ′ ) as well. This is acontradiction, since we can observe that no two distinct sequences in Table 7have three common α -successors. ✷ Proof of Theorem 16.
Let ( H ) be a minimal counterexample which is min-imal with respect to (cid:22) . By Lemma 17, ( H ) does not contain Configurations C C
8. There remains to show that every triangle-free planar graph con-tains at least one these 8 configurations. This has been already done using adischarging procedure in the proof of Theorem 2 in [8], where slightly weakerconfigurations were used. ✷ We now exhibit the following lower bound for the signified chromatic numberof planar graphs:
Theorem 18
There exist planar graphs with signified chromatic number . Proof.
Let ( G , Σ ) be the 6-path abcdef with Σ = { bc, de } and let ( G , Σ )be the 6-path abcdef with Σ = { ab, cd, ef } . Note that χ ( G , Σ ) = χ ( G , Σ )= 4.Let ( G ) be the outerplanar graph obtained from ( G , Σ ), ( G , Σ ) and avertex u such that u is positively (resp. negatively) linked to the six verticesof ( G ) (resp. ( G )). For any signified coloring of ( G ), we need 4 colors forthe vertices of ( G ), 4 other colors for the vertices of ( G ), and a ninth colorfor u . We therefore have χ ( G ) = 9.Let ( G ) be the graph obtained from two copies of ( G ) plus a vertex v suchthat v is positively linked to 13 vertices of the first copy of ( G ) and negativelylinked to the 13 vertices of the second one. This graph is depicted in Figure 8.Once again, it is easy to check that χ ( G ) = 19.Finally, let ( G ) be the graph obtained from 28 copies ( G ) , ( G ) , . . . , ( G ) of ( G ) as follows: we glue on each of the 27 vertices of ( G ) the vertex v of acopy ( G ) i . Since χ ( G ) = 19, we have χ ( G ) ≥
19. Suppose, χ ( G ) = 19.Therefore, there exists a graph ( H ) on 19 vertices such that ( G ) admits asignified homomorphism to ( H ). Moreover, since we glued a copy of ( G ) oneach vertices of ( G ) , then each color (i.e. each vertex of ( H )) must have 9distinct positive neighbors and 9 distinct negative neighbors. The subgraphof ( H ) induced by the positive edges is a 9-regular graph on 19 vertices. Sucha graph does not exist since, in every graph, the number of vertices of odddegree must be even. ✷ e c d a b i l k j g h i l k j g h f e c d a b u u f Fig. 8. A planar graph with signified chromatic number 19
Montejano et al. [6] proved that any outerplanar graph admits a signifiedhomomorphism to SP , that gives χ ( G ) ≤ G is an outerplanargraph. They also proved that this bound is tight. We prove here that SP isthe only suitable target graph on 9 vertices. Theorem 19
The only graph of order to which every outerplanar graphadmits a signified homomorphism is SP . Proof.
Let ( G ), ( G ) and ( G ) be the graphs constructed in the proof ofTheorem 18.Note that since χ ( G , Σ ) = 4, then for any signified 4-coloring of ( G , Σ ),among the three positive edges ab , cd and ef , two of them will use 4 distinctcolors. We will later refer to this property in the remainder of this proof asProperty 1. Using the same arguments as the previous paragraph, we havethat, for any signified 4-coloring of ( G , Σ ), among the three negative edges ab , cd and ef , two of them will use 4 distinct colors. We will later refer to thisproperty in the remainder of this proof as Property 2.Let ( G ′ ) be the outerplanar graph obtained from 14 copies ( G ) , ( G ) , . . . , ( G ) of ( G ) as follows: we glue on each of the 13 vertices of ( G ) the ver-tex u of a copy ( G ) i . Since χ ( G ) ≤ G is outerplanar, we have χ ( G ′ ) = 9. Therefore, there exists a graph ( H ) on 9 vertices such that ( G ′ )admits a signified homomorphism to ( H ). Each of the nine colors appears onthe vertices of the copy ( G ) . Since we glued a copy of ( G ) on each vertexof ( G ) , then each color c (i.e. each vertex c of ( H )) must have 4 positiveneighbors c +1 , c +2 , c +3 , c +4 and 4 negative neighbors c − , c − , c − , c − . Moreover, byProperty 1 (resp. Property 2), we must have a positive (resp. negative) match-ing in the subgraph induced by c +1 , c +2 , c +3 , c +4 (resp. c − , c − , c − , c − ).Meringer [5] provided an efficient algorithm to generate regular graphs witha given number of vertices and vertex degree. In particular, there exist 1618 a) (b) (c) (d)(e) (f) (g) (h)(i) (j) (k) ( ℓ )(m) (n) (o) (p)Fig. 9. The 16 connected 4-regular graphs on 9 vertices. connected 4-regular graphs on 9 vertices (see Figure 9). Replacing edges bypositive edges and non-edges by negative edges, these are the 16 signifiedgraphs such that each vertices have 4 positive and 4 negative neighbors. It isthen easy to check that, among these 16 graphs, there is only one graph witha positive (resp. negative) matching in the subgraph induced by the positive(resp. negative) neighbors of each vertex (see Figure 9(n)): this is SP . ✷ We finally give the same kind of result as Theorem 19 for planar graphs:
Theorem 20
If every planar graphs admits a signified homomorphism to ananti-twinned graph H of order , then H is isomorphic to T r ( SP ) . Proof.
Consider the outerplanar graph ( G ′ ) of the proof of Theorem 19. Let19 G ′ ) be the planar graph obtained from ( G ′ ) plus a universal vertex positivelylinked to all the vertices of ( G ′ ).The subgraph of ( G ′ ) induced by the vertices of ( G ′ ) necessarily maps to thepositive neighborhood of some vertex v of ( H ). Since ( H ) is anti-twinned,every vertex has exactly 9 positive neighbors. Therefore, by Theorem 19, thepositive neighborhood of v is isomorphic to SP . Then the subgraph of ( H )induced by v and its positive neighborhood is SP +9 . Since SP +9 is a cliqueof order 10, it does not contain a pair of anti-twin vertices and thus ( H ) isisomorphic to AT ( SP +9 ). Then, by definition of Tromp signified Paley graphs,( H ) is isomorphic to T r ( SP ). ✷ The upper bounds on signified chromatic number given in Theorems 12, 14, 15and 16 are obtained by showing that the considered graph class admits asignified homomorphism to some target graph. As mentioned in Corollary 4(1),when the target graph of a signified homomorphism is an anti-twinned graph,this gives a bound on the signed chromatic number. The upper bounds onsigned chromatic number given in this section are a direct consequence oftheses above-mentioned results.Naserasr et al. [7] proved the following:
Theorem 21 ([7])
Let G be a graph that admits an acyclic k -coloring. Wehave χ s ( G ) ≤ l k m · k − . By Theorem 12, ( G, Σ) admits a signified homomorphism to ZS k whenever G admits an acyclic k -coloring. By Proposition 5 and Corollary 4(1), we get thefollowing new upper bound that improves Theorem 21. Theorem 22
Let G be a graph that admits an acyclic k -coloring. We have χ s ( G ) ≤ k · k − . Recall that Huemer et al. [4] proved that Theorem 12 is tight, i.e. there existsa planar graph ( G ) such that χ ( G ) = k · k − . By Lemma 3, we can deducethat χ s ( G ) ≥ k · k − , showing that Theorem 22 is actually tight.We also get the following upper bounds: Theorem 23 (1) χ s ( O ) ≤ .(2) χ s ( P ) ≤ .(3) χ s ( P ) ≤ .(4) χ s ( P ) ≤ .(5) χ s ( P ) ≤ . Proof. (1) The result follows from Theorem 15 and Corollary 4(1).(2) By Theorem 12 and Borodin’s result [2], every planar graph admits asignified homomorphism to ZS , a graph on 80 vertices, which is anti-20winned by Proposition 5. The result then follows from Corollary 4(1).(3) The result follows from Theorem 16 and Corollary 4(1).(4) The result follows from Theorem 14(1) and Corollary 4(1).(5) The result follows from Theorem 14(2) and Corollary 4(1). ✷ Concerning lower bounds, Naserasr et al. [7] constructed a planar graph withsigned chromatic number 10. Note that this result also follows from The-orem 18 and Corollary 4(1). If 10 is the tight bound for signed chromaticnumber of planar graphs, we then get the following from Theorem 20 andLemma 3:
Theorem 24
If every planar graphs admits a signed homomorphism to agraph H of order , then H is isomorphic to SP +9 . We can easily construct a planar graph of girth 4 with signed chromatic num-ber 6 (see Figure 10). Finally, for higher girths, note that every even cyclewith exactly one negative edge needs 4 colors for any signed coloring.
One of our aims was to introduce and study some relevant target graphs forsignified homomorphisms. We studied the anti-twinned graph AT ( G, Σ), thesignified Zielonka graph ZS k , the signified Paley graph SP q , and the signifiedTromp Paley graph T r ( SP q ). Theorems 19, 20 and 24 suggest that such targetgraphs are indeed significant.We proved that there exist planar graphs with signified chromatic number 20and ask whether this bound is tight: Open Problem 25
Does every planar graph admit a signified homomorphismto
T r ( SP ) ? We have checked by computer that every 4-connected planar triangulationwith at most 15 vertices admits a homomorphism to
T r ( SP ). The restrictionto 4-connected triangulations (i.e. triangulations without separating triangles)is justified by Lemma 8. For the 2 non-equivalent signatures of each of the6244 4-connected planar triangulations with 15 vertices, our computer checktook 150 CPU-days. Checking 4-connected triangulations with more verticeswould require too much computing power.Finally, it would be nice to drop the condition “anti-twinned” in Theorem 20. References [1] N. Alon and T. H Marshall. Homomorphisms of edge-colored graphs and coxetergroups.
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