The mod k chromatic index of graphs is O(k)
aa r X i v : . [ m a t h . C O ] J u l The mod k chromatic index of graphs is O ( k ) ∗ F´abio Botler, † Lucas Colucci and Yoshiharu Kohayakawa ‡ Abstract
Let χ ′ k ( G ) denote the minimum number of colors needed to color the edges of a graph G in away that the subgraph spanned by the edges of each color has all degrees congruent to 1 (mod k ).Scott [ Discrete Math. 175 , 1-3 (1997), 289–291] proved that χ ′ k ( G ) ≤ k log k , and thus settled aquestion of Pyber [ Sets, graphs and numbers (1992), pp. 583–610], who had asked whether χ ′ k ( G )can be bounded solely as a function of k . We prove that χ ′ k ( G ) = O ( k ), answering affirmatively aquestion of Scott. Throughout this paper, unless stated otherwise, k ≥ e ( G ) denotes the number of edges in the graph G . A χ ′ k -coloring of G is a coloring ofthe edges of G in which the subgraph spanned by the edges of each color has all degrees congruent to1 (mod k ), and we denote by χ ′ k ( G ) the minimum number of colors in a χ ′ k -coloring of G . Pyber [3]proved that χ ′ ( G ) ≤ G and asked whether χ ′ k ( G ) is bounded by some function of k only. Scott [4] proved that χ ′ k ( G ) ≤ k log k for every graph G , and in turn asked if χ ′ k ( G ) is in factbounded by a linear function of k . This would be best possible apart from the multiplicative constant,as χ ′ k ( K ,k ) = k . In this paper, we answer Scott’s question affirmatively.We shall make use of the following two results. Lemma 1 (Mader [2]) . If k ≥ , G is a graph on n vertices, and e ( G ) ≥ kn , then G contains a k -connected subgraph. Lemma 2 (Thomassen [5]) . If k ≥ and G is a (12 k − -edge-connected graph with an even numberof vertices, then G has a spanning subgraph in which each vertex has degree congruent to k (mod 2 k ) . We say that a graph G is k -divisible if k divides the degree of every vertex of G . Lemma 2 thusguarantees the existence of a k -divisible spanning subgraph in G when G is (12 k − Lemma 3. If G is a graph on n vertices and does not contain a non-empty k -divisible subgraph, then e ( G ) < k − n .Proof. Let G be a graph and suppose that e ( G ) ≥ k − n . Lemma 1 tells us that G contains a(12 k − H . If H has an odd number of vertices, let H ′ = H − v for an arbitrary ∗ This research has been partially supported by Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil – CAPES– Finance Code 001. F. Botler is supported by CNPq (423395/2018-1) and by FAPERJ (211.305/2019). Y. Kohayakawa is partiallysupported by CNPq (311412/2018-1, 423833/2018-9) and FAPESP (2018/04876-1). The research that led to this paper started atWoPOCA 2019, which was financed by FAPESP (2015/11937-9) and CNPq (425340/2016-3, 423833/2018-9). FAPERJ is the Riode Janeiro Research Foundation. FAPESP is the S˜ao Paulo Research Foundation. CNPq is the National Council for Scientific andTechnological Development of Brazil. † Programa de Engenharia de Sistemas e Computa¸c˜ao, COPPE, Universidade Federal do Rio de Janeiro, Brazil ‡ Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Brazil v ∈ V ( H ). Otherwise, let H ′ = H . Then H ′ has an even number of vertices and is (12 k − k − H ′ contains a non-empty k -divisible subgraph and therefore so does G .Given an integer d , we say that a graph G is d -degenerate if there is an ordering v , . . . , v n of itsvertices for which the number of neighbors of v i in V i = { v , . . . , v i − } is at most d . In this case, theneighbors of v i in V i are its left neighbors and the neighbors of v i in { v i +1 , . . . , v n } are its right neighbors .An edge uv is a left edge of v if u is a left neighbor of v . A left edge uv of v is a right edge of u . Lemma 4.
Let G be a d -degenerate graph. Then χ ′ k ( G ) ≤ d + 2 k − .Proof. Let V ( G ) = { v , . . . , v n } be an ordering of V ( G ) as above. In what follows, we color G by coloringthe right edges of v i for each i ∈ { , . . . , n − } in turn, so that at each step we have a χ ′ k -coloring of thegraph spanned by the right edges of v , . . . , v i with the following properties: for each 1 ≤ j ≤ n , each ofthe colored, left edges of v j is colored with a distinct color, and the colors used on the right edges of v j are distinct from the colors used on its left edges. We proceed by induction on i . Let i ∈ { , . . . , n − } ,and suppose that, for every j < i , the right edges of v j is colored as above. This implies that all the leftedges of v i are colored and no right edge of v i is colored. In what follows, we color the set S of rightedges of v i while keeping the properties of the partial coloring.Since G is d -degenerate, v i has at most d left edges, and hence we have at least 3 d + 2 k − A and B so that | A | = d + k and | B | ≥ d + k −
2. Let j > i and suppose v j is a (right) neighbor of v i . Note that at most d − v j are colored. We say that a color c is forbidden at v j if a left edge of v j is colored with c , andwe call the colors in A that are not forbidden at v j available at v j .Let S ∗ be a maximal subset of S that can be colored with colors in A in a way that (a) each rightedge v i v j ∈ S ∗ is colored with a color available at v j , and (b) the number of edges in S ∗ colored withany given color is congruent to 1 (mod k ). Let ¯ S = S \ S ∗ be the set of the remaining edges in S . Weclaim that | ¯ S | < | A | . Assume for a contradiction that | ¯ S | ≥ | A | . For each edge e = v i v j ∈ ¯ S , let A e bethe set of colors available at v j , and for each color x ∈ A , let ¯ S x be the set of edges e in ¯ S for which x ∈ A e . Note that P e ∈ ¯ S | A e | = P x ∈ A | ¯ S x | and that | A e | ≥ | A | − ( d −
1) = d + k − d + 1 = k + 1 forevery e ∈ ¯ S . Therefore( k + 1) | A | ≤ ( k + 1) | ¯ S | ≤ X e ∈ ¯ S | A e | = X x ∈ A | ¯ S x | ≤ | A | max {| ¯ S x | : x ∈ A } , whence max {| ¯ S x | : x ∈ A } ≥ k + 1. Let z ∈ A be such that | ¯ S z | = max {| ¯ S x | : c ∈ A } . If some edge in S ∗ is colored with z , then we color k edges in ¯ S with color z . If no edge in S ∗ is colored with z , then wecolor k + 1 edges in ¯ S with color z . In both cases we obtain a contradiction to the maximality of S ∗ .This shows that, indeed, | ¯ S | < | A | = d + k .Finally, we color the edges of ¯ S consecutively and with distinct colors in B . This is possible, sincefor each v i w ∈ ¯ S there are at most d − | ¯ S | − ≤ d + k − < | B | colors of B that are forbidden (thecolors forbidden at w plus the colors of B used on previous edges of ¯ S ).Our main result is a consequence of Lemmas 3 and 4. Theorem 5.
For every graph G we have χ ′ k ( G ) ≤ k − .Proof. Let H be a maximal subgraph of G for which deg H ( v ) ≡ k ) for every v ∈ V ( H ), andlet G ′ = G \ E ( H ). The maximality of H implies that V ( G ) \ V ( H ) is independent, and that every2ertex in V ( H ) has at most k − V ( G ) \ V ( H ). Moreover, G ′ [ V ( H )] has no non-empty k -divisible subgraph. By Lemma 3, every J ⊆ G ′ [ V ( H )] has less than 2(12 k − | V ( J ) | edges, and henceits minimum degree is less than 48 k −
24. This implies that every subgraph of G ′ has a vertex of degreeat most 49 k −
25, and hence G ′ is (49 k − G ′ has a χ ′ k -coloringwith at most 198 k −
102 colors. We then color E ( H ) with a new color, and the result follows.Alon, Friedland and Kalai [1] proved that when k is a prime power, the bound 2(12 k − n in Lemma 3can be replaced by ( k − n + 1, and conjectured that this holds for every positive integer k . This result,together with Lemma 4, implies that χ ′ k ( G ) ≤ k − G and any prime power k .Our arguments can be tweaked to give slightly better multiplicative constants, but we do not thinkit is worth pursuing this because we think we would be far from the truth still. Although we are notable to offer any strong evidence, we conjecture the following. Conjecture 6.
There is a constant C such that χ ′ k ( G ) ≤ k + C for every graph G . We know that C in the conjecture above has to be at least 2, because one can prove that the graph G obtained from a K k,k by adding a universal vertex satisfies χ ′ k ( G ) = k + 2. References [1]
Alon, N., Friedland, S., and Kalai, G.
Regular subgraphs of almost regular graphs.
J. Combin.Theory Ser. B 37 , 1 (1984), 79–91.[2]
Mader, W.
Existenz n -fach zusammenh¨angender Teilgraphen in Graphen gen¨ugend grosser Kan-tendichte. Abh. Math. Sem. Univ. Hamburg 37 (1972), 86–97.[3]
Pyber, L.
Covering the edges of a graph by . . . . In
Sets, graphs and numbers (Budapest, 1991) (1992), vol. 60 of
Colloq. Math. Soc. J´anos Bolyai , North-Holland, Amsterdam, pp. 583–610.[4]
Scott, A. D.
On graph decompositions modulo k . Discrete Math. 175 , 1-3 (1997), 289–291.[5]
Thomassen, C.