On restricted colorings of $(d,s)$-edge colorable graphs

A cycle is $2$-colored if its edges are properly colored by two distinct colors. A $(d,s)$-edge colorable graph $G$ is a $d$-regular graph that admits a proper $d$-edge coloring in which every edge of $G$ is in at least $s-1$ $2$-colored $4$-cycles. Given a $(d,s)$-edge colorable graph $G$ and a list assigment $L$ of forbidden colors for the edges of $G$ satisfying certain sparsity conditions, we prove that there is a proper $d$-edge coloring of $G$ that avoids $L$, that is, a proper edge coloring $\varphi$ of $G$ such that $\varphi(e) \notin L(e)$ for every edge $e$ of $G$.


Introduction
A graph G is k-edge list colorable or k-edge choosable if for every assignment of lists of at least k colors to the edges of G, there is a proper edge coloring of G using only colors from the lists. The list chromatic index or edge choosability χ ′ l (G) of a graph G is the minimum number k such that G is k-edge list colorable. The most famous conjecture about list coloring states that χ ′ l (G) = χ ′ (G) [1], where χ ′ (G) is the chromatic index of G, refering to the smallest number of colors needed to color the edges of G to obtain a proper edge coloring. In 1994, Galvin [2] proved this conjecture for bipartite multigraphs, his result also answers a question of Dinitz in 1979 about the generalization of Latin squares which can be formulated as a result of list edge coloring of the complete bipartite graph K d,d , that χ ′ l (K d,d ) = d. Meanwhile, Häggkvist [3] worked with sparser lists, his conjecture of avoiding arrays can be rewritten in the language of graph theory to state that there exists a fixed 0 < β ≤ 1 3 such that if each edge e of K d,d is assigned a list L(e) of at most βd colors from {1, . . . , d} and at every vertex v each color is forbidden on at most βd edges adjacent to v, then there is a proper d-edge coloring ϕ of K d,d that avoids the lists, i.e ϕ(e) / ∈ L(e) for every edge e of K d,d ; if such a coloring exists, then L is avoidable. For the case when d is a power of two, Andrén proved that such a β exists in [4]; the full conjecture was later settled in the affirmative in [6].
Casselgren et al. [5] demonstrated that a similar result holds for the family of hypercube graphs. A benefit of working with the complete bipartite graph K d,d (d = 2 t , t ∈ N) and the d-dimensional hypercube graph Q d (d ∈ N) is that they are both regular graphs that have proper edge colorings in which every edge is in (d − 1) 2-colored 4-cycles. The purpose of this paper is to study this type of problem for regular graphs where the number of 2-colored 4-cycles each edge is contained in can be smaller.
To be more specific, we consider the family of (d, s)-edge colorable graphs: a d-regular graph G is called (d, s)-edge colorable if it admits a proper d-edge coloring h in which every edge of G is contained in at least (s − 1) 2-colored 4-cycles. Note that since the number of 2-colored 4-cycles containing an edge e of G is at most (d − 1), s can not exceed d. The d-edge coloring h is called standard coloring. A standard matching M of G is a maximum set of edges of G all of which have the same color in the standard coloring h.
The distance between two edges e and e ′ is the number of edges in a shortest path between an endpoint of e and an endpoint of e ′ ; a distance-t matching is a matching where any two edges are at distance at least t from each other. The t-neighborhood of an edge e is the graph induced by all edges of distance at most t from e.
Throughout, we shall assume that the standard coloring h for the edges of G uses the set of colors {1, . . . , d}. Next, similarly to [5], using the colors {1, . . . , d} we define a list assignment for the edges of a (d, s)-edge colorable graph.
A list assignment L for a (d, s)-edge colorable graph G is β-sparse if the list of each edge is a (possibly empty) subset of {1, . . . , d}, and (i) |L(e)| ≤ βs for each edge e ∈ G; (ii) for every vertex v ∈ V (G), each color in {1, . . . , d} occurs in at most βs lists of edges incident to v; (iii) for every 6-neighborhood W , and every standard matching M, any color appears at most βs times in lists of edges of M contained in W .
We can now formulate our main result.
Theorem 1.1. Let G be a (d, s)-edge colorable graph of order n and L be a β-sparse list assignment for G. If s ≥ 11 and β ≤ 2 −11 sd −1 (2n) −2 9 ds −2 , then there is a proper d-edge coloring of G which avoids L.
The following corollary gives us a shortened form of the condition on β as a function of d and n when s/d is constant. Corollary 1.2. Let G be a (d, s)-edge colorable graph of order n such that s ≥ 11 and s/d is constant; then there are positive constants c 1 , c 2 such that if β ≤ c 1 (2n) −c 2 d −1 , then any β-sparse list assignment L for G is avoidable.
If the value of d in Corollary 1.2 is at least c log n (c constant), then β will be constant. Corollary 1.3. Let G is a (d, s)-edge colorable graph of order n such that s ≥ 11 and suppose that there are constants κ ≤ 1 and c satisfying s = κd and d ≥ c log n; then there is a constant β > 0 such that any β-sparse list assignment L for G is avoidable.
Note that the complete bipartite graph K d,d (d = 2 t , t ∈ N) and the d-dimensional hypercube graph Q d (d ∈ N) are both (d, d)-edge colorable graphs satisfying the condition in Corollary 1.3. Thus this corollary generalizes the results in [4,5,6]. The next corollary examines the condition on d and s so that for every (d, s)-edge colorable graph G of order n and any β-sparse list assignment L for G satisfying that the length of every list in L is constant, there is a proper d-edge coloring of G which avoids L. Corollary 1.4. Let G be a (d, s)-edge colorable graph G of order n. If s ≥ 11 and for some constant c we have 1 2 (2 −11 c −1 s 2 d −1 ) 2 −9 d −1 s 2 ≥ n, then any c s -sparse list assignment L for G is avoidable.
If the length of every list in L is bounded by a power of s, we have a slightly different condition on d and s. Corollary 1.5. Let G be a (d, s)-edge colorable graph G of order n. If s ≥ 11 and for some constant c we have 1 2 (2 −11 s 2−c d −1 ) 2 −9 d −1 s 2 ≥ n, then any s c−1 -sparse list assignment L for G is avoidable.
Consider an arbitrary list assignment L ′ (not necessariliy β-sparse) for a (d, s)-edge colorable graph. In general, it is difficult to determine if L ′ is avoidable or not. However, if the edges with forbidden lists are placed on a distance-3 matching, our method in fact immediately yields the following. Theorem 1.6. Let L ′ be a list assignment for the edges of a (d, s)-edge colorable graph G such that for each edge e of G, L ′ (e) ≤ s − 1. If every edge e satisfying L ′ (e) = ∅ belongs to a distance-3 matching in G, then L ′ is avoidable.
The rest of the paper is organized as follows. In Section 2, after introducing some terminology and notation, we prove Theorem 1.1. Our proof relies heavily on the fact that every edge is contained in a large number of 2-colored 4-cycles. It would be interesting to investigate if a similar result holds for graphs containing a certain amount of 2-colored 2c-cycles (c ∈ N, c > 2). Section 3 gives some examples of classes of graphs that belong to the family of (d, s)-edge colorable graphs.

Proof the main theorem
Given a (d, s)-edge colorable graph G of order n, let h be a standard coloring of G. For a vertex u ∈ G, we denote by E u the set of edges with one endpoint being u, and for a (partial) edge coloring f of G, let f (u) denote the set of colors on edges in E u under f . If two edges uv and zt of G are in a 2-colored 4-cycle in G then the edges uv and zt are parallel.
Given a proper coloring h ′ of the edges of G, for an edge e ∈ G, any edge e ′ ∈ G (e ′ = e) belongs to at most one 2-colored 4-cycle containing e. This property is obvious if e ′ and e are not adjacent; in the case when they have the same endpoint u, assume e = uv, e ′ = uv ′ and uvv 1 v ′ u and uvv 2 v ′ u are two 2-colored 4-cycles containing e and e ′ ; then h ′ (vv 1 ) = h ′ (vv 2 ) = h ′ (uv ′ ), a contradiction since vv 1 and vv 2 are adjacent.
Consider a β-sparse list assignment L for G and a proper edge coloring φ of G. An edge e of G is called a conflict edge (of φ with respect to L) if φ(e) ∈ L(e). An allowed cycle (under φ with respect to L) of G is a 4-cycle C = uvztu in G that is 2-colored under φ, and such that interchanging colors on C yields a proper d-edge coloring φ ′ of G where none of uv, vz, zt, tu is a conflict edge. We call such an interchange a swap in φ.
We shall establish that our main theorem holds by proving two lemmas. In the following, G is a (d, s)-edge colorable graph of order n, L is a β-sparse list assignment for G, and h is a standard coloring of G. For simplicity of notation, we shall omit floor and ceiling signs whenever these are not crucial.
Lemma 2.1. Let 0 < γ, τ < 1 be parameters such that β ≤ γ and There is a permutation ρ of {1, . . . , d}, such that applying ρ to the set of colors {1, . . . , d} used in h, we obtain a proper d-edge coloring h ′ of G satisfying the following: (a) For every 6-neighborhood W , and every standard matching M, at most γs edges of M ∩ E(W ) are conflict.
(b) No vertex u in G satisfies that E u contains more than γs conflict edges.
(c) Each edge in G belongs to at least (1 − τ )s allowed cycles.
Proof. Let A, B, C be the number of permutations which do not fulfill the conditions (a), (b), (c), respectively. Let X be the number of permutations satisfying the three conditions (a), (b), (c). There are d! ways to permute the colors, so we have We will now prove that X is greater than 0.
• Since all edges that are in the same standard matching have the same color under h and for every 6-neighborhood W , and every standard matching M, any color appears at most βs times in lists of edges of M contained in W , we have that the maximum number of conflict edges in a subset of a given standard matching contained in a 6neighborhood is βs. Since γ ≥ β, this means that all permutations satisfy condition (a) or A = 0.
• To estimate B, let u be a fixed vertex of G, and let P be a set of size γs (|P | = γs) of edges from E u . For a vertex v adjacent to u, if uv is a conflict edge, then the colors used in h should be permuted in such a way that in the resulting coloring h ′ , the color of uv is in L(uv). Since |L(uv)| ≤ βs, there are at most (βs) γs ways to choose which colors from {1, 2, . . . , d} to assign to the edges in P so that all edges in P are conflict. The rest of the colors can be arranged in any of the (d − γs)! possible ways. In total this gives at most d γs permutations that do not satisfy condition (b) on vertex u. There are n vertices in G, so we have B ≤ n d!(βs) γs (γs)!
• To estimate C, let uv be a fixed edge of G. Each 2-colored 4-cycle C = uvztu containing uv is uniquely defined by an edge zt which is parallel with uv. Moreover, a permutation ς contributes to C if and only if there are at least τ s choices for zt so that C is not allowed. We shall count the number of ways ς could be constructed for this to happen. First, note that for each choice of a color c 1 from {1, . . . , d}, for the standard matching which contains uv, there are up to 2βs cycles that are not allowed because of this choice. This follows from the fact that there are at most βs choices for t (or z) such that L(ut) (or L(vz)) contains c 1 . So for a permutation ς to contribute to C, ς must satisfy that at least (τ − 2β)s cycles containing uv are forbidden because of the color assigned to the standard matching containing ut and vz.
Let S be a set of edges, |S| = (τ − 2β)s, such that for every edge zt ∈ S, the 2-colored 4-cycle C = uvztu is not allowed because of the color assigned to ut and vz. There are s−1 (τ −2β)s ways to choose S. Furthermore, L(uv) and L(zt) contain at most βs colors each, so there are at most 2βs choices for a color for the standard matching containing ut and vz that would make C disallowed because of the color assigned to this standard matching. The remaining colors can be permuted in (d − 1 − (τ − 2β)s)! ways. Thus, the total number of permutations σ with not enough allowed cycles for a given edge is bounded from above by Notice that the d-regular graph G of order n has nd 2 edges, so the total number of permutations σ that have too few allowed cycles for at least one edge is bounded from above by Hence,

Using the conditions
we now deduce that X > 0.
Lemma 2.2. Let h ′ be the proper d-edge coloring satisfying conditions (a), (b), (c) of Lemma 2.1 and 0 < γ, τ, ǫ < 1 be parameters such that By performing a sequence of swaps on disjoint allowed 2-colored 4-cycles in h ′ , we obtain a proper d-edge coloring h ′′ of G which avoids L.
Proof. For constructing h ′′ from h ′ , we will perform a number of swaps on G, and we shall refer to this procedure as P-swap. We are going to construct a set P of disjoint allowed 2-colored 4-cycles such that each conflict of h ′ with L belongs to one of them. An edge that belongs to a 2-colored 4-cycle in P is called used in P -swap. Suppose we have included a 4-cycle C in P . Since for every 6-neighborhood W in G, and every standard matching M, the number of conflict edges in M ∩ E(W ) is not greater than γs, for every 5-neighborhood W in G, the total number of edges in W that are used in P -swap is at most 4γds. A vertex u in G is P -overloaded if E u contains at least ǫs edges that are used in P -swap; note that each used edge is incident to two vertices, thus no more than 2 × 4γds ǫs = 8γ ǫ d vertices of each 4-neighborhood are P -overloaded. A standard matching M in G is P -overloaded in a t-neighborhood W if M ∩ E(W ) contains at least ǫs edges that are used in P -swap; note that for each 5-neighborhood W , no more than 4γ ǫ d standard matchings of G are P -overloaded in W .
Using these facts, let us now construct our set P by steps; at each step we consider a conflict edge e and include an allowed 2-colored 4-cycle containing e in P . Initially, the set P is empty. Next, for each conflict edge e = uv in G, there are at least s − τ s allowed cycles containing e. We choose an allowed cycle uvztu which contains e and satisfies the following: (1) z and t and the standard matching that contains vz and ut are not P -overloaded in the 4-neighborhood W e of e; this eliminates at most Note that with this strategy for including 4-cycles in P , after completing the construction of P , every vertex is incident with at most 2γs+(ǫs−1)+2 = 2γs+ǫs+1 edges that are used in P -swap. Furthermore, after we have constructed the set P , no standard matching contains more than 2γs + ǫs + 1 edges that are used in S-swap in a 1-neighborhood of G ; this follows from the fact that every 1-neighborhood W ′ in G that ut, vz or zt belongs to is contained in W e .
(2) None of the edges vz, zt, ut are conflict, or used before in P -swap.
All possible choices for these edges are in the 1-neighborhood W e of e in G . Since no vertex in W e or subset of a standard matching that is in W e contains more than γs conflict edges and P -swap uses at most 2γs + ǫs + 1 edges at each vertex and in each subset of a standard matching contained in W e , these restrictions eliminate at most 3γs + 3(2γs + ǫs + 1) or 9γs + 3ǫs + 3 choices.
It follows that we have at least choices for an allowed cycle uvztu which contains uv. By assumption, this expression is greater than zero, so we conclude that there is a cycle satisfying these conditions, and thus we may construct the set P by iteratively adding disjoint allowed 2-colored 4-cycles such that each cycle contains a conflict edge. After this process terminates we have a set P of disjoint allowed cycles; we swap on all the cycles in P to obtain the coloring h ′′ which avoids L.

Families of (d, s)-edge colorable graphs
The results in [4] and [5] allow us to conclude that the complete bipartite graph K d,d (d = 2 t , t ∈ N) and the hypercube graph Q d (d ∈ N) are (d, d)-edge colorable graphs. Compared to the hypercube graph Q d (d ∈ N), the complete bipartite graph K d,d (d = 2 t , t ∈ N) is much denser, so it is interesting to see how this complete bipartite graph behaves if some edges are removed. Lemma 3.1 considers the case when we take the set of k standard matchings out of K d,d (d = 2 t , t ∈ N).
Lemma 3.1. The graph G obtained by removing k standard matchings from the complete bipartite graph Proof. It is straightforward that G is a (d − k)-regular graph. Let h be a standard coloring of K d,d such that all edges in a standard matching of K d,d receive the same color in h. We define the proper (d − k)-edge coloring h ′ of G from h by retaining the color of every non-deleted edge in E(G). Consider an edge e of G, removing one standard matching from K d,d eliminates one 2-colored 4-cycle that contains e. Hence, the number of 2-colored 4-cycles containing e Recall that the Cartesian product G = G 1 G 2 of the graphs G 1 and G 2 is a graph whose vertex set is the Cartesian product V (G 1 ) × V (G 2 ) and where two vertices u = (u 1 , u 2 ) and v = (v 1 , v 2 ) are adjacent in G whenever u 1 = v 1 and u 2 is adjacent with v 2 in G 2 , or u 2 = v 2 and u 1 is adjacent with v 1 in G 1 . The following lemma yields (d, s)-edge colorable graphs by taking Cartesian product of graphs.
Lemma 3.2. Let G 1 be a (d 1 , s 1 )-edge colorable graph and G 2 be a (d 2 , s 2 )-edge colorable graph. Then the Cartesian product G = G 1 G 2 of graphs G 1 and G 2 is a (d, s)-edge colorable graph with d = d 1 + d 2 and s = min{d 1 + s 2 , d 2 + s 1 }.
Proof. By the definition of the Cartesian product of graphs, it is straightforward that G is d-regular graph with d = d 1 + d 2 . Let h 1 be a standard coloring of G 1 and h 2 be a standard coloring of G 2 such that the set of colors in h 1 and the set of colors in h 2 are disjoint. We define an edge coloring h of G: For two adjacent vertices u = (u 1 , u 2 ) and v = (u 1 , v 2 ) in G, the edge uv is given the color h(uv) = h 2 (u 2 v 2 ) and for two adjacent vertices u = (u 1 , u 2 ) and v = (v 1 , u 2 ) in G, the edge uv is given the color h(uv) = h 1 (u 1 v 1 ). Thus h is a proper d-edge coloring of G.
Note that an edge uv of G with u = (u 1 , u 2 ) and v = (u 1 , v 2 ) is contained in a 2-colored 4-cycle uvztu with z = (u i , v 2 ) and t = (u i , u 2 ) (u i is neighbour of u 1 in G 1 ). Furthermore, if u 2 v 2 z 2 t 2 u 2 is a 2-colored 4-cycle in G 2 , then uvz ′ t ′ u with z ′ = (u 1 , z 2 ) and t ′ = (u 1 , t 2 ) is a 2-colored 4-cycle in G. Since the degree of u 1 is d 1 and every edge of G 2 is in at least s 2 − 1 2-colored 4-cycles, an edge uv of G with u = (u 1 , u 2 ) and v = (u 1 , v 2 ) belongs to at least d 1 + s 2 − 1 2-colored 4-cycles. Similarly, an edge uv of G with u = (u 1 , u 2 ) and v = (v 1 , u 2 ) belongs to at least d 2 + s 1 − 1 2-colored 4-cycles. Therefore, we can conclude that G is a (d, s)-edge colorable graph with d = d 1 + d 2 and s = min{d 1 + s 2 , d 2 + s 1 }.
In the remaining part of this section, we examine some other graphs that belong to the family of (d, s)-edge colorable graphs.
Let G be a finite group and let S be a generating set of G such that S does not contain the identity element e, |S| = d and S = S −1 (which means if a ∈ S then a −1 ∈ S). The undirected Cayley graph Cay(G, S) over the set S is defined as the graph whose vertex set is G and where two vertices a, b ∈ G are adjacent whenever {ab −1 , ba −1 } ⊆ S. It is straightforward that Cay(G, S) is a d-regular graph, Lemma 3.3 and Lemma 3.4 show that if S satisfies some further conditions then Cay(G, S) is a (d, s)-edge colorable graph.  Proof. We write S = {s 1 , . . . , s k , s −1 1 , . . . , s −1 k }; note that the size of S may not be 2k, since S may contain some element s x with s x = s −1 x . Consider an edge uv of Cay(G, S); without loss of generality assume that v = us i (u = vs −1 i ) for some i ∈ [1, k]. The condition (ii) implies that there exists exactly one sequence (x 1 , x 2 , . . . , x k ) such that u = s 1 Given an edge e of Cay(G, S), let u and v be the two endpoints of e, where v = us i (for some i ∈ [1, k]). Let u = s 1 x 1 ...s i x i ...s k x k ; x i is called the power of s i in u (i ∈ [1, k]) and denoted by p u (s i ). Consider an arbitrary element s ∈ S, if s = s j ∈ S k , let z = vs j , t = us j , then h(vz) = h(ut) = s j or h(vz) = h(ut) = s −1 j since p u (s j ) = p v (s j ) and p t (s j ) = p z (s j ) = p u (s j ) + 1. Furthermore, since G is an Abelian group, we have z = vs j = us i s j = us j s i = ts i . Thus there is an edge between z and t, and h(uv) = h(tz) since p u (s i ) = p t (s i ) and p v (s i ) = p z (s i ) = p u (s i ) + 1; hence uvztu is a 2-colored 4-cycle. If s ∈ S −1 k , we proceed similarly. Because |S| = d, each edge of Cay(G,S) is in at least d − 1 2-colored 4-cycles. It follows that Cay(G, S) is a (d, d)-edge colorable graph.

Acknowledgement
The author would like to thank Klas Markström and Carl Johan Casselgren for their help and useful discussion.