Regular colorings and factors of regular graphs
Anton Bernshteyn, Omid Khormali, Ryan R. Martin, Jonathan Rollin, Danny Rorabaugh, Songling Shan, Andrew J. Uzzell
aa r X i v : . [ m a t h . C O ] M a r Regular colorings and factors of regular graphs ✩ Anton Bernshteyn a,1 , Omid Khormali b , Ryan R. Martin c,2 , Jonathan Rollin d ,Danny Rorabaugh e , Songling Shan f , Andrew J. Uzzell g a Department of Mathematics, University of Illinois at Urbana-Champaign b Department of Mathematical Sciences, University of Montana c Department of Mathematics, Iowa State University d Department of Mathematics, Karlsruhe Institute of Technology e Department of Mathematics and Statistics, Queen’s University f Department of Mathematics, Vanderbilt University g Department of Mathematics, University of Nebraska–Lincoln
Abstract
An ( r − , -coloring of an r -regular graph G is an edge coloring such that eachvertex is incident to r − , { r − , } -factor of an r -regular graph is a spanning subgraph in which eachvertex has degree either r − , { , } -factors. Finally, for each r ≥ r − , r − t, t )-colorable for small t . Keywords: r -regular graph, { r − , } -factor, ( r − ,
1. Introduction
A graph with no loops or multiple edges is called simple ; a graph in whichboth multiple edges and loops are allowed is called a pseudograph . Unless speci-fied otherwise, the word “graph” in this paper is reserved for pseudographs. All(pseudo)graphs considered here are undirected and finite. Note that we counta loop twice in the degree of a vertex.The famous Berge–Sauer conjecture asserts that every 4-regular simple graphcontains a 3-regular subgraph [6]. This conjecture was settled by Tashkinov in1982 [11]. In fact, he proved that every connected 4-regular pseudograph witheither at most two pairs of multiple edges and no loops or at most one pair ofmultiple edges and at most one loop contains a 3-regular subgraph. Observe that ✩ This research was partially supported by NSF Grant 1500662 “The 2015 Rocky Mountain- Great Plains Graduate Research Workshop in Combinatorics.” Research of this author is supported by the Illinois Distinguished Fellowship. This work was supported by a grant from the Simons Foundation (
Preprint his cannot hold for all 4-regular pseudographs, because the graph consisting ofa single vertex with two loops contains no 3-regular subgraph. The followingquestion remains open.
Question 1.1.
Which 4-regular pseudographs contain 3-regular subgraphs?Note that in 1988, Tashkinov [12] classified the values of t and r for whichevery r -regular pseudograph contains a t -regular subgraph. Beyond findingregular subgraphs in regular graphs, finding factors—that is, regular spanningsubgraphs—in regular graphs is also of special interest. As early as 1891, Pe-tersen [9] studied the existence of factors in regular graphs. Since then numerousresults on factors have appeared—see, for example, [2, 5, 7, 10]. The concept offactors can be generalized as follows: for any set of integers S , an S -factor of agraph is a spanning subgraph in which the degree of each vertex is in S . Severalauthors [1, 3, 8] have recently studied { a, b } -factors in r -regular graphs with a + b = r . In particular, Akbari and Kano [1] made the following conjecture: Conjecture 1.2. If r is odd and 0 ≤ t ≤ r , then every r -regular graph has an { r − t, t } -factor.However, Axenovich and Rollin [3] disproved this conjecture. The followingtheorem summarizes what is known about { r − t, t } -factors of r -regular graphs.(Note that although intended for simple graphs, the result of Petersen [9] appliesto pseudographs as well.) Theorem 1.3.
Let t and r be positive integers with t ≤ r .(a) When r is even: • If t is even, then every r -regular graph has a t -factor, and thus has an { r − t, t } -factor (Petersen [9]). • Every r -regular graph of even order has an (cid:8) r + 1 , r − (cid:9) -factor (Lu,Wang, and Yu [8]). • If t is odd and t ≤ r − , then there exists an r -regular graph of evenorder that has no { r − t, t } -factor ([8]). • If t is odd, then trivially, no r -regular graph of odd order has an { r − t, t } -factor.(b) When r is odd and r ≥ : • If t is even, then every r -regular graph has an { r − t, t } -factor (Akbariand Kano [1]). • If t is odd and r ≤ t , then every r -regular graph has an { r − t, t } -factor([1]). • If t is odd and ( t + 1)( t + 2) ≤ r , then there exists an r -regular graphthat has no { r − t, t } -factor (Axenovich and Rollin [3]).(c) Every -regular graph has a { , } -factor (Tutte [14]). r = 5 and t = 1. As we will give much of our attention to this case, we restateit separately. Conjecture 1.4.
Every 5-regular graph has a { , } -factor.An ( r − t, t ) -coloring of an r -regular graph G is an edge-coloring (with atleast two colors) such that each vertex is incident to r − t edges of one color and t edges of a different color. An ordered ( r − t, t ) -coloring of G is an ( r − t, t )-coloringusing integers as colors such that each vertex is incident to r − t edges of somecolor i and t edges of some color j with i < j . Bernshteyn [4] introduced (3 , , Theorem 1.5 (Bernshteyn [4]) . A connected -regular graph contains a -regular subgraph if and only if it admits an ordered (3 , -coloring. We observe that the notion of an ( r − t, t )-coloring of an r -regular graphgeneralizes that of an { r − t, t } -factor, because { r − t, t } -factors correspond to ( r − t, t )-colorings that use exactly two colors. (In an r -regular graph with 0 < t < r , t -factors correspond to ordered ( r − t, t )-colorings that use exactly two colors.)Thus, ( r − t, t )-colorings provide a common approach to attacking Question 1.1and Conjecture 1.4. This leads us to ask whether the following weaker versionof Conjecture 1.4 holds. Question 1.6.
Does every 5-regular graph have a (4 , r ≥
6, the answer to the analogue of Question 1.6 for ( r − , Question 1.7.
Which 4-regular graphs have (3 , t -factors, { r − t, t } -factors, ordered ( r − t, t )-colorings, ( r − t, t )-colorings, and t -regular subgraphsof r -regular graphs.Now we are ready to describe our main results. First, in Section 2, wecharacterize all 4-regular graphs which are not (3 , , { , } -factors. Finally, in Section 4, we construct relevant examples of r -regulargraphs for r ≥ t : some with no ( r − t, t )-coloring, others with an( r − t, t )-coloring but no { r − t, t } -factor.3 has a t -factor. G has an { r − t, t } -factor. G has an ordered ( r − t, t )-coloring. G has a t -regular subgraph. G has an ( r − t, t )-coloring. Figure 1: Implications that hold for every r -regular graph G and for all integers 0 < t < r .
2. (3 , In this section, we characterize 4-regular graphs that do not admit (3 , G and G be vertex-disjointgraphs with edges e = u v ∈ E ( G ) and e = u v ∈ E ( G ). The edgeadhesion of G and G at e and e is the graph G = ( G , e ) + ( G , e )obtained by subdividing edges e and e and identifying the two new vertices.(See Figure 2.) That is, V ( G ) = V ( G ) ˙ ∪ V ( G ) ˙ ∪ { w } ; E ( G ) = ( E ( G ) \ { e } ) ˙ ∪ ( E ( G ) \ { e } ) ˙ ∪ { u w, v w, u w, v w } .e + e G G = G Figure 2: Edge adhesion of two graphs, G = ( G , e ) + ( G , e ). The adhesion of a loop to graph H at edge e = uv ∈ E ( H ) is the graph H ′ = ( H, e ) + O obtained by subdividing e and adding a loop at the new vertex.(See Figure 3.) That is, V ( H ′ ) = V ( H ) ˙ ∪ { x } ; E ( H ′ ) = ( E ( H ) \ { e } ) ˙ ∪ { ux, vx, xx } .eH + O = H ′ Figure 3: Adhesion of a loop at an edge, H ′ = ( H, e ) + O . C be a (simple) cycle. A double cycle is obtained from C by doublingeach edge. We say a double cycle is even (respectively, odd) if it has an even(respectively, odd) number of vertices. (See Figure 4.) · · ·· · · Figure 4: Double cycles (odd on top, even on bottom).
Clearly, double cycles and graphs resulting from edge adhesion of two 4-regular graphs or from the adhesion of a loop to a 4-regular graph are all 4-regular. We are now ready to give the main result of this section.
Theorem 2.1.
A connected -regular graph is not (3 , -colorable if and only ifit can be constructed from odd double cycles via a sequence of edge adhesions.Remark . Theorem 2.1 naturally lends itself to a proof by induction. Inparticular, an equivalent statement is that a connected 4-regular graph is not(3 , , Lemma 2.3.
A double cycle with n ≥ vertices is (3 , -colorable if and onlyif n is even.Proof. Even double cycles have perfect matchings and are thus (3 , , c of an odd double cycle G . Let G ′ denote the cycle obtained by removing one of the parallel edges between anytwo adjacent vertices in G . Color an edge in G ′ red if the corresponding edgesin G are of the same color under c and blue otherwise. Observe that the edgesincident to any vertex in G ′ are of different colors, since c is a (3 , G . This is a contradiction since G ′ is an odd cycle. Lemma 2.4 (Bernshteyn [4]) . If G is a -regular graph and there exists a non-double edge uv in G with u = v such that G − { u, v } is connected, then G is (3 , -colorable. Lemma 2.5 (Bernshteyn [4]) . If G is a -regular graph and G ′ = ( G, e ) + O for some edge e ∈ E ( G ) , then either G or G ′ has a -regular subgraph. Lemma 2.6.
Let G and G be (3 , -colorable -regular graphs and let G havea loop vv . Construct G by subdividing an edge uw in G , identifying the newvertex with v , and removing the loop vv , so V ( G ) = V ( G ) ˙ ∪ V ( G ); E ( G ) = ( E ( G ) \ { uw } ) ˙ ∪ ( E ( G ) \ { vv } ) ˙ ∪ { uv, wv } . (See Figure 5.) Then G is (3 , -colorable. wG G v −→ G uwv
Figure 5: Joining G to G at a loop, as in Lemma 2.6. Proof.
Fix (3 , c i of G i for i ∈ { , } . Note that v in G is incidentto only one loop and that the two non-loop edges incident to v have differentcolors under c . Without loss of generality, assume that c ( uw ) is equal to thecolor of one of the non-loop edges incident to v . Therefore the colorings c and c extend to a (3 , G by coloring the edges uv and uw withcolor c ( uw ). Corollary 2.7 (to Lemmas 2.5, 2.6) . Suppose exactly one of the connected -regular graphs G and G is (3 , -colorable. Then for any e ∈ E ( G ) and e ∈ E ( G ) , ( G , e ) + ( G , e ) is (3 , -colorable.Proof. Without loss of generality, we assume that G is (3 , G is not. Let e ∈ E ( G ) and e ∈ E ( G ). By Theorem 1.5 and Lemma 2.5, thegraph G ′ = ( G , e ) + O is (3 , G and G ′ ,we see that ( G , e ) + ( G , e ) is (3 , Lemma 2.8.
Let G be a -regular graph that is not (3 , -colorable. If G has anon-double, non-loop edge, then G is not -connected.Proof. Let uv be a non-double, non-loop edge, and suppose for contradictionthat G is 2-connected. By Lemma 2.4, since G is not (3 , G ′ = G − { u, v } is disconnected. Since G is 2-connected, neither u nor v is a cut-vertex. Therefore, every component of G ′ must contain at least one vertex from N G ( u ) and at least one vertex from N G ( v ). Since the sum of the degrees of thevertices must be even in each component, the 4-regularity of G implies that eachcomponent of G ′ must have an even number of vertices from N G ( u ) ∪ N G ( v ).Let N G ( u ) \ { v } = { u , u , u } and N G ( v ) \ { u } = { v , v , v } . Without loss ofgenerality, G ′ is the disjoint union of a component G containing u and v anda subgraph G (of one or two components) containing u , u , v , and v .Let G ′ = ( G + u v , u v ) + O and G ′ = (( G − G ) + uv, uv ) + O . (SeeFigure 6.) That is, V ( G ′ ) = V ( G ) ˙ ∪ { w } ; E ( G ′ ) = E ( G ) ˙ ∪ { u w , v w , w w } ; V ( G ′ ) = V ( G ) ˙ ∪ { u, v, w } ; E ( G ′ ) = E ( G ) ˙ ∪ { uu , uu , uv, vv , vv , uw , vw , w w } . By the assumption of 2-connectedness, the vertex u is not a cut-vertex of G , so u = v and G ′ − { u , w } is connected. Thus by Lemma 2.4, G ′ is6 u vu v G u v u v G w u u v vu v G ′ u v w G ′ Figure 6: Splitting a 2-connected graph into two (3 , (3 , G ′ − { u, w } is connected, so G ′ is (3 , , c i of G ′ i for i ∈ { , } . Note that because of the loops, c ( u w ) = c ( v w ) and c ( uw ) = c ( vw ). We can assume that c ( u w ) = c ( uw ) and c ( v w ) = c ( vw ). Therefore, the colorings c and c easilyextend to a (3 , c of G , which is a contradiction. Lemma 2.9.
Let G be a connected -regular graph that is not -connected. Then G = ( G , e ) + ( G , e ) for some -regular graphs G , G and edges e ∈ E ( G ) , e ∈ E ( G ) .Proof. Indeed, let w ∈ V ( G ) be a cut-vertex. Now the lemma is implied bythe following observation. Since the number of vertices with odd degrees in agraph is always even, G − w consists of exactly two components and each ofthese components receives exactly two of the edges incident to w . Proof of Theorem 2.1.
Consider 4-regular graphs G and G and edges e in G , e in G . Any (3 , G , e ) + ( G , e ) yields a (3 , G or G , since the edges obtained by subdividing e or e are of the samecolor. Therefore every graph that is obtained from odd double cycles via edgeadhesion is not (3 , G be a connected 4-regular graph that is not (3 , | V ( G ) | to prove that G is constructed from odd double cyclesvia edge adhesion. If | V ( G ) | = 1, then G is a double cycle of one vertex andthe theorem trivially holds. Assume that | V ( G ) | ≥
2. We may also assume that G contains a non-double edge. Otherwise, if every edge is double, then G is adouble cycle, and by Lemma 2.3, G is an odd double cycle, and thus we aredone.If each non-double edge is a loop, then one can easily check that G is not2-connected. If G has a non-double non-loop edge, Lemma 2.8 implies that itis not 2-connected. By Lemma 2.9, G = ( G , e ) + ( G , e ) for some 4-regular7raphs G , G and edges e ∈ E ( G ), e ∈ E ( G ). Corollary 2.7 implies thateither both G and G are (3 , , G and G are (3 , G ′ = ( G , e )+ O andobserve that G is obtained from G ′ and G as in the statement of Lemma 2.6.Since G is (3 , G is not, Lemma 2.6 implies that G ′ is not(3 , G ′ is obtained fromodd double cycles via edge adhesion. Since G ′ contains a loop and at least twovertices, it is not a double cycle. Thus, G ′ = ( G ′ , e ′ ) + ( G ′ , e ′ ), where both G ′ and G ′ are not (3 , G ′ does not contain the subdivided edge e , and so G = ( G ′ , e ′ ) + ( H, f ) forsome graph H and edge f in H . Since both G and G ′ are not (3 , H by Corollary 2.7. We have shown that G is obtained from twographs that are not (3 ,
3. (4 , { , } -factors in 5-regular graphs In this section, we make progress toward settling Conjecture 1.4 and Ques-tion 1.6. In particular, we show that if G is a vertex-minimal counterexampleto Conjecture 1.4, then G must satisfy a large number of structural conditions.We show that similar conditions must hold for any vertex-minimal graph thatgives a negative answer to Question 1.6.A set S of edges of a connected graph G is called an edge cut if G − S isdisconnected. An edge cut S is minimal provided G − ( S \ { e } ) is connected foreach edge e ∈ S . An edge cut of size 1 is called a bridge . Note that a minimaledge cut does not contain loops.Most of the following results are obtained using reductions to smaller graphs.We also use a corollary of Tutte’s 1-Factor Theorem. Theorem 3.1 (Tutte [13]) . A graph G has a -factor if and only if the numberof connected components of G − S of odd order is at most | S | for every vertexset S ⊆ V ( G ) . Corollary 3.2.
Every k -edge-connected (2 k + 1) -regular graph has a -factor. In Section 3.1, we prove our results about (4 , { , } -factors. -regular graphs without (4 , -colorings We begin by showing that a vertex-minimal 5-regular graph with no (4 , c of G extends an edge-coloring c ′ of G ′ if c ( e ) = c ′ ( e ) for all e ∈ E ( G ) ∩ E ( G ′ ). Theorem 3.3.
Let G be a vertex-minimal -regular graph without a (4 , -coloring.(a) G is connected. b) G is not -edge-connected, i.e., contains an edge cut on edges.(c) G has no minimal edge cut of size .(d) G does not have two bridges.(e) Each bridge in G has (precisely) one endpoint incident to two loops.(f ) The edges of any minimal edge cut of size in G have a vertex in common,and this vertex is incident to a loop.Proof. (a) This follows from vertex-minimality.(b) This is a consequence of Corollary 3.2 with k = 2.(c) Assume { uv, wx } is a minimal edge cut, so G − { uv, wx } is disconnected,but G − uv and G − wx are both connected. Then G −{ uv, wx } has precisely twocomponents, G and G , and, without loss of generality, u , w ∈ V ( G ) and v , x ∈ V ( G ). We obtain 5-regular graphs G ′ = G + uw and G ′ = G + vx byadding a new edge (possibly a loop or parallel edge) to each component. By theassumption of vertex-minimality, both graphs G ′ and G ′ have (4 , c i of G ′ i for i ∈ { , } such that c ( uw ) = c ( vx ) = 1.Note that all edges of E ( G ) \ { uv, wx } are contained in exactly one of G ′ or G ′ . So we obtain a (4 , G by coloring uv and wx with color 1 andall other edges according to c and c , a contradiction.(d) Assume uv and wx are bridges in G . Then G − { uv, wx } has threecomponents. Without loss of generality, assume that u and w are contained inthe same component. We obtain two 5-regular graphs by adding the edges uw and vx (possibly loops or parallel edges). The proof proceeds exactly as in (c).(e) If there is a bridge with both endpoints incident to two loops, then thereare no other edges and the graph is easily (4 , uv with each endpoint incident to at most one loop. Then G − uv hastwo components G and G , each with at least 2 vertices. We obtain a 5-regulargraph from G (respectively, G ) by adding a new vertex incident to two loopsand to u (respectively, to v ). Both graphs have (4 , , G by choosingthe same color for the new edges incident to u and v , a contradiction.(f) Consider distinct edges uv , wx and yz forming a minimal edge cut ofsize 3. First observe that a vertex which is incident to all three edges is incidentto a loop due to statements (c) and (d) of this theorem. Thus assume thatthere is no such vertex. Removing the three edges from G yields exactly twocomponents G and G , each with at least 2 vertices. Without loss of generalityassume u , w and y are in G and v , x and z are in G .Let H i denote the set of all 5-regular graphs that contain G i as a subgraphand have one more vertex than G i , i ∈ { , } . Note that H i = ∅ . We considereach H ∈ H i with a fixed copy K = K ( H ) of G i and call edges in E ( H ) \ E ( K ) new if they are incident to vertices of K . By assumption all graphs in H ∪ H , i ∈ { , } , there is a graph H ∈ H i having a (4 , , G , a contradiction.So, assume that for any graph H ∈ H and for any (4 , H thenew edges in H are not all of the same color. Consider a (4 , c of thegraph in H obtained from G by adding a new vertex p incident to one loopand connected to u , w and y by new edges. Without loss of generality assumethat c ( up ) = c ( wp ) = c ( yp ). Further consider a (4 , c of the graphin H obtained from G by adding by adding a new vertex q incident to twoloops and edges vx and qz . Then c ( vx ) = c ( qz ) by assumption. Therefore weobtain a (4 , G from c and c as before, a contradiction.Now we prove a number of conditions involving loops, parallel edges, orforbidden subgraphs (see Figure 7). Theorem 3.4.
Let G be a vertex-minimal -regular graph without a (4 , -coloring.(a) G does not contain a -regular subgraph with at least vertices.(b) G does not have parallel edges.(c) G does not contain a path of length three consisting of double edges.(d) No vertex of G that has a loop is incident to a double edge.(e) No vertices with loops are adjacent.(f ) G contains at least loops.(g) There do not exist u , u , u , v , v , v ∈ V ( G ) such that the u i have loopsand such that for each i and j , u i is adjacent to v j (that is, there is no K , with one loop on each vertex of one side of the vertex partition).(h) No vertex is adjacent to more than vertices with loops.(i) No -vertex subgraph of G has or more edges. Figure 7: From Theorem 3.4 (b, c, d, e, g, h), forbidden subgraphs in a vertex-minimal5-regular graph with no (4 , roof. (a) Suppose for contradiction that G has a 4-regular subgraph H withat least 2 vertices. Let F denote the set of edges uv in G with u ∈ V ( H ) and v V ( H ). We obtain a 5-regular graph G ′ from G by removing the verticesof H and adding some new edges between the vertices of degree less than 5 and,if there is an odd number of such vertices, one new vertex with two loops. Let F ′ denote the set of new edges in G ′ , except for the loops incident to the newvertex, if such exists.By vertex-minimality, G ′ has a (4 , c . We extend this to a coloringof G as follows. Assign a color k not used by c to all edges in H and a colordifferent from k to all edges in E ( G ) \ E ( H ) having both endpoints in H . Eachedge in F shares a vertex with least one edge in F ′ . Consider an injective map f : F → F ′ such that e and f ( e ) have a common vertex for all e ∈ F . Thencolor each e ∈ F with color c ( f ( e )). This coloring is a (4 , G , acontradiction.(b) Assume that there are at least three edges between vertices u and v .Let F denote the set of edges incident to u or v but not both. Observe that G has at least 3 vertices, as the 2-vertex 5-regular graphs are easily (4 , G ′ obtained from G by removing u and v and adding a matching between the (remaining) neighborhood of u and the(remaining) neighborhood of v , possibly creating parallel edges and loops. Byassumption G ′ has a (4 , c . We extend this to a coloring of G bycoloring the edges in F with the colors of the corresponding new edges under c . Then u and v are either both incident to edges of the same color only, orboth incident to an edge of one color and an edge of a second color. In eithercase we can color the parallel edges between u and v such that we obtain a(4 , G , a contradiction.(c) Let v , v , v , and v denote the vertices of a double path in G . Let u be the other neighbor of v and u the other neighbor of v . We assumethat u = v and u = v due to part (b) of this Theorem. Remove v and v from G , add two edges between v and v , and add an edge (possibly a loop ormultiple edge) between u and u . Let G ′ denote the resulting graph, which,by hypothesis, has a (4 , v and v , as well as the edge u u , havecolor 1. Then in G , we give color 1 to all edges incident to v or v except forone of the edges between v and v , to which we give color 2.Second, suppose the edges between v and v have color 1 and the edge u u has color 2. Then in G , we give color 2 to u v and u v and color 1 to all otheredges incident to v or v .Third, suppose u u and one of the edges between v and v have color 1,while the other has color 2. Then in G we give color 2 to one of the edgesbetween v and v and to one of the edges between v and v . We give color 1to all other edges incident to v or v .Fourth, suppose all three edges have different colors. Then there are twosubcases to consider. If one of the edges between v and v is the only edge ofits color that is incident to both v and v , then we may instead give it the same11 igure 8: Extending the coloring of G ′ to G in the proof of Theorem 3.4 (c). color as u u and so reduce the problem to the previous case. Assume, then,that the edges between v and v have colors 1 and 2, that v is incident to fouredges with color 1, that v is incident to four edges with color 2, and that u u has color 3. When we color G , we give color 1 to one of the edges between v and v , color 2 to one of the edges between v and v , and color 3 to all otheredges incident to v or v .We have shown that in all four cases, we may extend a (4 , G ′ to a (4 , G , which is a contradiction.(d) Suppose to the contrary that u is a vertex with a loop and that there isa double edge between u and some other vertex v . Observe that v cannot havea loop: if it did, then u and v would each have exactly one neighbor outsideof { u, v } . This is a minimal edge cut of size 2, which contradicts Theorem 3.3 (c).Thus, u sends one edge to a vertex w outside of { u, v } , while v sends three,to vertices x , y , and z . We remove u and v from G and create a 5-regulargraph G ′ by adding edges e = wx and f = yz . (As usual, we may create loopsor multiple edges.) By hypothesis, G ′ has a (4 , e and f both havecolor 1, then we may extend the coloring to G by coloring all edges incident to u or to v with color 1, except for one edge between u and v , to which we givecolor 2. If e has color 1 and f has color 2, then we extend the coloring to G bygiving color 1 to both uw and vx and color 2 to all other edges incident to u or v . In either case, we have a contradiction.(e) Suppose that u and v are adjacent vertices with loops. By part (d), thereis exactly one edge between u and v . Observe that neither u nor v can have12wo loops. Indeed, if both have two loops, then G [ { u, v } ] is a component andobviously has a (4 , v has two loops but u has only one, then u sends two edges to vertices outside of { u, v } . These edges form a minimaledge cut of size 2, which contradicts Theorem 3.3 (c).Thus, we may assume that u and v are incident to only one loop each andhence both send two edges to vertices outside of { u, v } . Delete u and v andform a new 5-regular graph G ′ by adding a matching between the (remaining)neighborhood of u and the (remaining) neighborhood of v . By hypothesis, G ′ has a (4 , , G much as in part (b),a contradiction.(f) If X and Y are disjoint subsets of V ( G ), let e ( X, Y ) denote the numberof edges between X and Y . Since G does not admit a (4 , S ⊂ V ( G )such that the number of components of G − S of odd order is strictly greaterthan | S | . Let C , . . . , C t be the components of G − S of of odd order. Then5 | V ( C i ) | = 2 | E ( C i ) | + e ( S, C i ) and hence e ( S, C i ) is odd, 1 ≤ i ≤ t . Similarly,there is an even number of edges between S and a component of G − S of evenorder. Therefore | S | ≡ t (mod 2), and thus t ≥ | S | + 2.Recall from Theorem 3.3 (e, f) that if G contains a bridge, then one of theendpoints of the bridge is incident to two loops, and if G contains a minimaledge cut of size 3, then its edges share an endpoint which is incident to a loop.Therefore, either e ( S, C i ) ≥ C i and this vertex isadjacent to ℓ ≥ e ( S, C i ) + 2 ℓ = 5. Let C = S ti =1 C i and let k be the total number of loops in G . Then5 t ≤ (cid:12)(cid:12) e ( S, C ) (cid:12)(cid:12) + 2 k ≤ | S | + 2 k and hence k ≥
52 ( t − | S | ) ≥ · , as desired.(g) By part (a), we may assume that the v i form an independent set, becauseif, say, v v were an edge in G , then { v , v , u , u , u } would induce a 4-regularsubgraph of G . Delete all of the u i and the v i and form a new 5-regular graph G ′ by adding a matching M = { e , e , e } among the neighborhoods of the v i such that each edge e ij (which may be a loop) has one endpoint in N ( v i ) andthe other in N ( v j ).By hypothesis, G ′ has a (4 , c . When we extend this coloring to G , we will give c ( e ij ) to one edge incident to v i and to one edge incident to v j .Furthermore, we will give to each vertex v i an ordered triple ( a , a , a ) :=( c ( v i u ) , c ( v i u ) , c ( v i u )). There are three cases we must consider (see Figure 9).First, suppose that all of the e ij have color 1. In this case, we give v thetriple (2 , , v the triple (1 , , v the triple (1 , , u i .Second, suppose that the e ij have exactly two colors. Without loss of gen-erality, let c ( e ) = c ( e ) = 1 and c ( e ) = 2. Observe that in G , v is incident13 igure 9: Extending the coloring of G ′ to G in the proof of Theorem 3.4 (g). to two edges with color 1, while v and v are each incident to one edge withcolor 1 and one edge with color 2. We give the triple (1 , ,
2) to v , (2 , , v , and (1 , ,
1) to v . Additionally, we give color 1 to the loops at u and u and color 2 to the loop at u .Third, suppose that all of the e ij have different colors. Without loss ofgenerality, let c ( e ) = 1, c ( e ) = 2, and c ( e ) = 3. In this case, we give thetriple (1 , ,
1) to v and (2 , ,
2) to both v and v . Additionally, we give color 2to the loop at each u i .In all three cases, we have produced a (4 , G , which is a con-tradiction.(h) Suppose that u ∈ V ( G ) is adjacent to vertices v , v , v , and v , all ofwhich have loops. By part (e), the v i form an independent set. By part (d),there is only one edge between u and each v i . Delete the v i and form a new5-regular graph G ′ by adding two loops at u , and, for each i such that v i hasonly one loop, adding an edge e i between the two vertices of N ( v i ) \ { u } . Byhypothesis, G ′ has a (4 , G as follows:for each v i with only one loop, we give all edges incident to v i , except for uv i ,the same color as e i . We then give each uv i the color of the loops incident to u in G ′ , which we may assume is a new color. Finally, if any of the v i have twoloops, we give these loops a color different from the color of uv i . Thus, G has a(4 , H be a subgraph of G with 4 vertices and 8 edges. Since G is 5-regular,there are 4 edges between H and G − H . Let U = { u, v, w, x } be the multiset ofvertices in H , where the multiplicity of a vertex in U equals the number of edgesbetween the vertex and G − H . Let U ′ = { u ′ , v ′ , w ′ , x ′ } be the correspondingneighbors in G − H . 14n each of the seven present graphs, H has a 1-factor M and there are twodistinct vertices in U , u and v (without loss of generality), so that H − { u, v } has a non-loop edge e . Define the 5-regular graph G ′ = ( G − H ) + { u ′ v ′ , w ′ x ′ } ,which has fewer vertices than G . By our assumption of vertex-minimality, G ′ has a (4 , c . We can then define a (4 , G as follows. Let c = c ( u ′ v ′ ) and c = c ( w ′ x ′ ). If c = c , use a new color for the edges in theone-factor M and use c for all other edges incident to a vertex in H . If c = c ,use c for uu ′ , vv ′ , and e and use c for all other edges incident to a vertex in H . vxuw vxu w uw vxuw vx u xv w xu v w u vw x Figure 10: 4-vertex, 8-edge graphs with the vertices in U labeled and an edge in H − { u, v } dashed, from the proof of Theorem 3.4 (i). -regular graphs without { , } -factors The results in this subsection are very similar to those in the previous sub-section, so we will omit some of the proofs. Notice first that every statement ofTheorem 3.3 also holds for vertex-minimal graphs without { , } -factors becausethe proofs do not require the use of more than two colors. Theorem 3.5.
Let G be a vertex-minimal -regular graph without a { , } -factor.(a) G is connected.(b) G is not -edge-connected, i.e., contains an edge cut on edges.(c) G has no minimal edge cut of size .(d) G does not have two bridges.(e) Each bridge in G has (precisely) one endpoint incident to two loops.(f ) The edges of any minimal edge cut of size in G have a vertex in common,and this vertex is incident to a loop. Most of the statements in Theorem 3.4 also hold for vertex-minimal graphswithout { , } -factors. We discuss the differences between Theorems 3.4 and 3.6in Remark 3.7 below. 15 heorem 3.6. Let G be a vertex-minimal -regular graph without a { , } -factor.(a) G does not contain a copy of K .(b) G has no parallel non-loop edges.(c) No vertices with loops are adjacent.(d) G contains at least 5 loops.(e) There do not exist u , u , u , v , v , v ∈ V ( G ) such that the u i have loopsand such that for each i and j , u i is adjacent to v j .Remark . Here, we elaborate on the relationships between the statements inTheorems 3.4 and 3.6. First, the proofs of Theorem 3.4 (a, h) do not work forfactors, since in each case, we may need three colors to create the contradictory(4 , G .Next, the proof of Theorem 3.6 (a) given below does not work for general(4 , , , G ′ , making ex-tension to a (4 , G impossible.) So, the analogous statements toTheorem 3.4 (c, d) for factors are merely special cases of forbidding parallelnon-loop edges. Similarly, with no parallel non-loop edges and, by Theorem 3.5(d, e), at most one double loop, the analogous statement to Theorem 3.4 (i) isimmediate.Finally, the proofs of Theorem 3.4 (e, f, g), which correspond to Theo-rem 3.6 (c, d, e), work for { , } -factors in exactly the same way, so we will notgive the proofs. Proof of Theorem 3.6. (a) Let K = { u , u , v , v } denote the vertices of a copyof K in G . We obtain a graph G ′ by removing all vertices in K from G andadding two adjacent new vertices u and v . Then, for each x / ∈ K , we add anedge xu for each edge xu i in G and an edge xv for each edge xv i in G , i ∈ { , } .(Note that this may create multiple edges.)The new graph G ′ is 5-regular and has fewer vertices than G . By the as-sumption of vertex-minimality, it has a { , } -factor. This { , } -factor extendsto a { , } -factor of G regardless of the colors of the edges incident to u and v (see Figure 11). This is a contradiction.16 igure 11: All possible configurations of a { , } -factor at the edge uv and corresponding { , } -factors using edges from the copy of K (up to taking complements of color classes),from the proof of Theorem 3.6 (a). (b) Assume that there are at least two edges between u and v . Considerthe 5-regular graph G ′ obtained from G by removing u and v and adding amatching between N ( u ) \ { v } and N ( v ) \ { u } (possibly creating parallel edgesand loops). By assumption, G ′ has a { , } -factor F . We can extend F to a { , } -factor of G by adding some of the edges between u and v to F , which isa contradiction. r -Regular Graphs for r ≥ In this section we give a negative answer to the analogue of Question 1.6 for r ≥
6. More generally, for each odd t and each even r , as well as for each odd t and each odd r ≥ ( t +2)( t +1), we construct an r -regular graph with no ( r − t, t )-coloring. Note that for even t , every r -regular graph has a ( r − t, t )-coloring andfor odd t ≤ r and even r every r -regular graph has a ( r − t, t )-coloring due toTheorem 1.3. Theorem 4.1.
Let r and t be positive integers with t ≤ r odd. If r is evenor r ≥ ( t + 2)( t + 1) , then there exists an r -regular graph that is not ( r − t, t ) -colorable. Observe that this is the same upper bound on odd r as in Theorem 1.3(b)(due to [3]) for the existence of r -regular graphs without { r − t, t } -factors. Proof.
First, if r is even, then the r -regular graph with one vertex and r loopshas no ( r − t, t )-coloring, since t is odd.Now suppose that r ≥ ( t + 2)( t + 1) ≥ G be a graph on vertices v , u , u , . . . , u t +1 with t + 2 edges between v and u i and r − t − loops incidentto u i , 1 ≤ i ≤ t + 1, and r − ( t + 2)( t + 1) ≥ v and u and ( t +2)( t +1)2 loops incident to u . Observe that G is r -regular. Suppose that G admits an ( r − t, t )-coloring and observe that in any such coloring, there is an17 such that all t + 2 edges between v and u i are of the same color. However,this is a contradiction, because there is no coloring of the loops incident to this u i such that there are exactly t edges of another color incident to u i , as t isodd.Now we will exhibit r -regular graphs of even order that have ( r − , { r − , } -factors. The constructions are similar to construc-tions in [8]. Theorem 4.2.
For every even r ≥ there exists an ( r − , -colorable r -regulargraph of even order without an { r − , } -factor.Proof. Note that K r +1 has an odd number of vertices and thus does not havean { r − , } -factor, as r − r − , K r in red, r − r is odd, then let G , . . . , G r be vertex-disjoint copies of K r +1 − e . Forma graph G from the union of G i by connecting all vertices of degree r − G i to a new vertex u . Then G has an even number of vertices and is r -regular.Moreover there is an ( r − , u and extend the coloring to each G i , 1 ≤ i ≤ r , using thecoloring of K r +1 given above. Assume that G has an { r − , } -factor, i.e., an( r − , i , 1 ≤ i ≤ r , such that bothedges between G i and u are of the same color. This yields an ( r − , K r +1 in two colors, a contradiction.If r is even, then let t = 3( r − G , . . . , G t be vertex-disjoint copiesof K r +1 − e . Form a graph G from the union of the G i and a disjoint copyof K with vertex set { u , u , u } by connecting both vertices of degree r − G i to u j if j ( r − < i ≤ ( j + 1)( r − G has an even number ofvertices and is r -regular. One can show that G has an ( r − , { r − , } -factor with arguments similar to those given above.
5. Concluding Remarks
Here we state a number of open problems related to our work. Recall fromthe Introduction that Tashkinov [11] showed that every 4-regular graph with nomultiple edges and at most one loop contains a 3-regular subgraph. It is notknown whether the restriction on the number of loops is necessary.
Question 5.1.
Does every 4-regular graph with no multiple edges have a 3-regular subgraph?Let us note that Question 5.1 is open even for the class of 4-regular graphswith no multiple edges and at most two loops.Our next question concerns ( r − , G is a 4-regular graph that has a (3 , G has a (3 , uestion 5.2. Is there a positive integer K such that every 5-regular graphhas a (4 , K colors?Question 5.2 lies “between” Conjecture 1.4 and Question 1.6 in the follow-ing sense. An affirmative answer to Question 5.2 clearly gives an affirmativeanswer to Question 1.6. On the other hand, as observed in the Introduction,Conjecture 1.4 implies an affirmative answer to Question 5.2 with K = 2. Letus also note that none of the proofs of the statements in Theorems 3.3 and 3.4required more than three colors.Our final question concerns ordered ( r − , Question 5.3.
For r ≥
5, if G is an r -regular graph with an ( r − G admit an ordered ( r − , r = 4. Acknowledgments
We are grateful to Maria Axenovich, Sogol Jahanbekam, Yunfang Tang,Claude Tardif, and Torsten Ueckerdt for helpful conversations.
References [1] S. Akbari and M. Kano. { k, r − k } -factors of r -regular graphs. GraphsCombin. , 30(4):821–826, 2014.[2] J. Akiyama and M. Kano.
Factors and factorizations of graphs: Prooftechniques in factor theory , volume 2031 of
Lecture Notes in Mathematics .Springer, Heidelberg, 2011.[3] M. Axenovich and J. Rollin. Brooks type results for conflict-free coloringsand { a, b } -factors in graphs. Discrete Math. , 338(12):2295–2301, 2015.[4] A. Yu. Bernshteyn. 3-regular subgraphs and (3 , J. Appl. Ind. Math. , 8(4):458–466, 2014.[5] B. Bollob´as, A. Saito, and N. C. Wormald. Regular factors of regulargraphs.
J. Graph Theory , 9(1):97–103, 1985.[6] J. A. Bondy and U. S. R. Murty.
Graph theory with applications . AmericanElsevier Publishing Co., Inc., New York, 1976.[7] F. R. K. Chung and R. L. Graham. Recent results in graph decompositions.In
Combinatorics (Swansea, 1981) , volume 52 of
London Math. Soc. LectureNote Ser. , pages 103–123. Cambridge Univ. Press, Cambridge-New York,1981. 198] H. Lu, D. G. L. Wang, and Q. Yu. On the existence of general factors inregular graphs.
SIAM J. Discrete Math. , 27(4):1862–1869, 2013.[9] J. Petersen. Die Theorie der regul¨aren graphs.
Acta Math. , 15(1):191–220,1891.[10] M. D. Plummer. Graph factors and factorization: 1985–2003: a survey.
Discrete Math. , 307(7–8):791–821, 2007.[11] V. A. Tashkinov. 3-regular subgraphs of 4-regular graphs.
Mat. Zametki ,36(2):239–259, 1984.[12] V. A. Tashkinov. Regular parts of regular pseudographs.
Mat. Zametki ,43(2):263–275, 1988.[13] W. T. Tutte. The factorization of linear graphs.
J. London Math. Soc. ,22:107–111, 1947.[14] W. T. Tutte. The subgraph problem.