On Restricted Colorings of \(\text{(d,s)}\)-Edge Colorable Graphs

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\(\chi '_l(G)\) 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 \(\chi '_l(G)=\chi '(G)\), where \(\chi '(G)\) 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 [1] proved this conjecture for bipartite multigraphs, his result also answers a question of Dinitz (1979) [2] about a generalization of Latin squares which can be formulated as a result of list edge coloring of the complete bipartite graph \(K_{d,d}\), that \(\chi '_l(K_{d,d})=d\).

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 \(0<\beta \le \frac{1}{3}\) such that if each edge e of \(K_{d,d}\) is assigned a list L(e) of at most \(\beta d\) colors from \(\{1, \dots ,d\}\) and at every vertex v each color is forbidden on at most \(\beta d\) edges adjacent to v, then there is a proper d-edge coloring \(\varphi\) of \(K_{d,d}\) that avoids the lists, i.e., \(\varphi (e) \notin L(e)\) 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 \(\beta\) exists in [4]; the full conjecture was later settled in the affirmative in [5].

Casselgren et al. [6] demonstrated that a similar result holds for the family of hypercube graphs. A benefit of working with the complete bipartite graph \(K_{d,d}\) (\(d=2^t\), \(t \in {\mathbb {N}}\)) and the d-dimensional hypercube graph \(Q_d\) (\(d \in {\mathbb {N}}\)) 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:

Definition 1

A d-regular graph G is called (d, s)-edge colorable if it admits a proper d-edge coloring in which every edge of G is contained in at least \((s-1)\) 2-colored 4-cycles.

Remark

For a (d, s)-edge colorable graph G, let \({\mathcal {H}}\) be the set of all proper d-edge colorings that satisfy the condition in Definition 1. Since \(|{\mathcal {H}}| \ge 1\), we can pick one proper d-edge coloring \(h \in {\mathcal {H}}\) (h can be chosen arbitrarily) and let it be our standard coloring. A standard matchingM 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 \(e'\). The t-neighborhood of an edge e is the graph induced by all edges of distance at most t from e.

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. Throughout, we shall assume that the standard coloring h for the edges of G uses the set of colors \(\{1,\dots , d\}\). Next, similarly to [6], using the colors \(\{1,\dots , d\}\) we define a \(\beta\)-sparse list assignment for the edges of a (d, s)-edge colorable graph.

Definition 2

A list assignment L for a (d, s)-edge colorable graph G is \(\beta\)-sparse if the list of each edge is a (possibly empty) subset of \(\{1,\dots ,d\}\), and

1. (i)

   \(|L(e)| \le \beta s\) for each edge \(e \in G\);

2. (ii)

   for every vertex \(v \in V(G)\), each color in \(\{1,\dots ,d\}\) occurs in at most \(\beta s\) lists of edges incident to v;

3. (iii)

   for every 6-neighborhood W of G, and every standard matching M of G, any color appears at most \(\beta s\) times in lists of edges of M contained in W.

The 6-neighborhood used in Definition 2 is solely due to proof technical reasons. The neighborhood size might be decreased, but this seems to be out of reach with current method. We can now formulate our main result.

Theorem 1

For any positive integersn, d, ssuch that\(s \ge 11\)and any positive\(\beta\)such that\(\beta \le 2^{-11}sd^{-1} (2n)^{-2^9ds^{-2}}\); ifGis a (d, s)-edge colorable graph of ordernthen any\(\beta\)-sparse list assignmentLforGis avoidable.

The rest of the paper is organized as follows. In Sect. 2, after introducing some terminology and notation, we prove Theorem 1. Section 3 contains several corollaries that are deduced directly from the proof of Theorem 1. In Sect. 4, we give some examples of classes of graphs that belong to the family of (d, s)-edge colorable graphs.

Given a (d, s)-edge colorable graph G of order n, for a vertex \(u \in G\), 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 \(e \in G\), any edge \(e' \in G\) (\(e' \ne e\)) 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 \(uvv_1v'u\) and \(uvv_2v'u\) 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 \(\beta\)-sparse list assignment L for G and a proper edge coloring \(\phi\) of G. An edge e of G is called a conflict edge (of \(\phi\)with respect toL) if \(\phi (e) \in L(e)\). An allowed cycle (under\(\phi\)with respect toL) of G is a 4-cycle \({\mathcal {C}}=uvztu\) in G that is 2-colored under \(\phi\), and such that interchanging colors on \({\mathcal {C}}\) yields a proper d-edge coloring \(\phi '\) of G where none of uv, vz, zt, tu is a conflict edge. We call such an interchange a swap in \(\phi\).

In the following, G is a (d, s)-edge colorable graph of order n, L is a \(\beta\)-sparse list assignment for G, and h is the standard coloring of G. For simplicity of notation, we shall omit floor and ceiling signs whenever these are not crucial. Below we outline the proof of Theorem 1.

1. Step I.

   We prove that there exists a permutation \(\rho\) of the elements of the set \(\{1,\dots ,d\}\) such that in the proper d-edge coloring \(h'\) obtained by applying \(\rho\) to the colors used in h, locally, each standard matching in G contains “sufficiently few” conflict edges with L, and each vertex u of G satisfies that \(E_u\) contains “sufficiently few” conflict edges, and that each edge of G belongs to “many” allowed cycles. These conditions shall be more precisely articulated below.

2. Step II.

   We find a set of allowed cycles in \(h'\) such that each conflict edge belongs to one of them, with no two of the cycles intersecting, and swap on those cycles to obtain a proper d-edge coloring \(h''\) of G which avoids L.

In the proof we shall verify that it is possible to perform Steps I–II described above to obtain a proper d-edge coloring of G that avoids L. This is done by proving a lemma in each step.

Step I We use Lemma 1 to prove that there exists a permutation \(\rho\) such that applying \(\rho\) to the colors used in h, we obtain a required proper d-edge coloring \(h'\).

Lemma 1

Let\(0<\gamma , \tau <1\)be parameters such that\(\beta \le \gamma\)and

Then there is a permutation\(\rho\)of\(\{1,\dots ,d\}\), such that applying\(\rho\)to the set of colors\(\{1,\dots ,d\}\)used inh, we obtain a properd-edge coloring\(h'\)ofGsatisfying the following:

1. (a)

   For every 6-neighborhoodWofG, and every standard matchingMofG, at most\(\gamma s\)edges of\(M \cap E(W)\)are conflict edges.

2. (b)

   For each vertexuinG, \(E_u\)contains at most\(\gamma s\)conflict edges.

3. (c)

   Each edge inGbelongs to at least\((1-\tau )s\)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.

Let’s start by counting A. Since all edges that are in the same standard matching have the same color under h and for every 6-neighborhood W of G, and every standard matching M of G, any color appears at most \(\beta s\) 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 6-neighborhood is \(\beta s\).

Since \(\gamma \ge \beta\), this means that all permutations satisfy condition (a); so \(A=0\).

To estimate B, let u be a fixed vertex of G, and let P be a set of size \(\gamma s\) (\(|P|=\gamma s\)) 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 \(|L(uv)| \le \beta s\), there are at most \((\beta s)^{\gamma s}\) ways to choose which colors from \(\{1,2,\dots ,d\}\) 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 \((d - \gamma s)!\) possible ways. In total this gives at most

permutations that do not satisfy condition (b) on vertex u. There are n vertices in G, so we have

To estimate C, let uv be a fixed edge of G. Each 2-colored 4-cycle \({\mathcal {C}} = uvztu\) containing uv is uniquely defined by an edge zt which is parallel with uv. Moreover, a permutation \(\varsigma\) contributes to C if and only if there are at least \(\tau s\) choices for zt so that \({\mathcal {C}}\) is not allowed. We shall count the number of ways \(\varsigma\) could be constructed for this to happen. First, note that for each choice of a color \(c_1\) from \(\{1,\dots , d\}\), for the standard matching which contains uv, there are up to \(2\beta s\) cycles that are not allowed because of this choice. This follows from the fact that there are at most \(\beta s\) choices for t (or z) such that L(ut) (or L(vz)) contains \(c_1\). So for a permutation \(\varsigma\) to contribute to C, \(\varsigma\) must satisfy that at least \((\tau -2\beta )s\) cycles containing uv are forbidden because of the color assigned to the standard matching containing ut and vz.

Let \({\mathcal {C}}_{uv}\) be the set of edges that are parallel with uv. It is obvious that \(s-1 \le |{\mathcal {C}}_{uv}| \le d-1\). Let \(S \subseteq {\mathcal {C}}_{uv}\), \(|S|=(\tau -2\beta ) s\), such that for every edge \(zt \in S\), the 2-colored 4-cycle \({\mathcal {C}}= uvztu\) is not allowed because of the color assigned to ut and vz. There are at most \({d-1} \atopwithdelims (){(\tau - 2\beta )s}\) ways to choose S. Furthermore, L(uv) and L(zt) contain at most \(\beta s\) colors each, so there are at most \(2\beta s\) choices for a color for the standard matching containing ut and vz that would make \({\mathcal {C}}\) disallowed because of the color assigned to this standard matching. The remaining colors can be permuted in \((d-1-(\tau -2\beta )s)!\) ways. Thus, the total number of permutations \(\sigma\) with not enough allowed cycles for a given edge is bounded from above by

Notice that any d-regular graph G of order n has \(\dfrac{nd}{2}\) edges, so the total number of permutations \(\sigma\) that have too few allowed cycles for at least one edge is bounded from above by

Hence,

By assumption, we now deduce that \(X>0\). \(\square\)

Step II We use Lemma 2 to prove 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.

Lemma 2

Let\(h'\)be a properd-edge coloring satisfying conditions (a), (b), (c) of Lemma 1and\(\gamma\), \(\tau\)be parameters satisfying condition of Lemma 1. If there exists\(\epsilon\)such that\(0<\epsilon <1\)and

Then, by performing a sequence of swaps on disjoint allowed 2-colored 4-cycles in\(h'\), we obtain a properd-edge coloring\(h''\)ofGwhich avoidsL.

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 2-colored 4-cycle \({\mathcal {C}}\) in P. Since for every 6-neighborhood W of G, and every standard matching M of G, the number of conflict edges in \(M \cap E(W)\) is not greater than \(\gamma s\), for every 5-neighborhood W of G, the total number of edges in W that are used in P-swap is at most \(4\gamma d s\). A vertex u in G is P-overloaded if \(E_u\) contains at least \(\epsilon s\) edges that are used in P-swap; note that each used edge is incident to two vertices, thus no more than \(\dfrac{2 \times 4\gamma d s}{\epsilon s}=\dfrac{8\gamma }{\epsilon }d\) vertices of each 4-neighborhood are P-overloaded. A standard matching M in G is P-overloaded in a t-neighborhood W if \(M \cap E(W)\) contains at least \(\epsilon s\) edges that are used in P-swap; note that for each 5-neighborhood W, no more than \(\dfrac{4\gamma }{\epsilon }d\) 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 \(s- \tau s\) allowed cycles containing e. We choose an allowed cycle uvztu which contains e and satisfies the following:

1. (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 \(\dfrac{2 \times 8\gamma }{\epsilon }d + \dfrac{4\gamma }{\epsilon }d=\dfrac{20\gamma }{\epsilon }d\) choices. Note that with this strategy for including 4-cycles in P, after completing the construction of P, every vertex is incident with at most \(2\gamma s+(\epsilon s -1) +2 = 2\gamma s+\epsilon s +1\) edges that are used in P-swap. Furthermore, after we have constructed the set P, no standard matching contains more than \(2\gamma s+\epsilon s+1\) 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. (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 \(\gamma s\) conflict edges and P-swap uses at most \(2\gamma s+\epsilon s +1\) edges at each vertex and in each subset of a standard matching contained in \(W_e\), these restrictions eliminate at most \(3 \gamma s + 3(2 \gamma s+\epsilon s +1)\) or \(9 \gamma s + 3\epsilon s+3\) 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 a coloring \(h''\) which avoids L. \(\square\)

We can now prove Theorem 1.

Proof of Theorem 1

Firstly, using Stirling’s approximation \(x! \ge x^xe^{-x}\), we have

By assumption, we have \(\beta \le 2^{-11}sd^{-1} (2n)^{-2^9ds^{-2}}\). Since \(s \le d\) and \((2n)^{-2^9ds^{-2}} \le 1\), it follows that \(\beta \le 2^{-11} sd^{-1} \le 2^{-11}\). Let \(\gamma =2^{-9}sd^{-1}\), then \(\beta \le \gamma\) and

Let \(\tau =2^{-7}\), then \(\tau -2\beta \ge 2^{-7} -2.2^{-11}>2^{-8}\) and

This implies

Using the fact that \(d <n\), we have

Combining (3) and (4), we obtain \(n\left( \dfrac{e \beta }{\gamma }\right) ^{\gamma s} +\dfrac{nd}{2}\left( \dfrac{2e \beta }{\tau -2\beta }\right) ^{(\tau -2\beta ) s} < 1\). This implies \(n \dfrac{(\beta s)^{\gamma s}}{(\gamma s)!} + \dfrac{nd}{2}\dfrac{{(2\beta s)^{(\tau - 2\beta )s}}}{((\tau - 2\beta )s)!} <1\). Since the values of \(\gamma , \tau\) satisfy the conditions in Lemma 1, there is a permutation \(\rho\) of the colors in the standard d-edge coloring h of G from which we obtain a proper d-edge coloring \(h'\) of G satisfying the conditions (a), (b), (c) in Lemma 1. Furthermore, since \(d\ge s \ge 11\), if we let \(\epsilon =2^{-3}\), then we have

It follows from Lemma 2 that there exits a proper d-edge coloring \(h''\) of G which avoids L.

Remark

1. (1)

   If \(2^{-11}sd^{-1} (2n)^{-2^9ds^{-2}} <1/s\), then \(\beta s <1\), Theorem 1 becomes obvious. So our result is only meaningful if \(2^{-11}sd^{-1} (2n)^{-2^9ds^{-2}} \geq 1/s\).

2. (2)

   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 (\(c \in {\mathbb {N}}\), \(c >2\)).

\(\square\)

The following corollaries are deduced directly from the proof of Theorem 1.

Corollary 1

For any positive integersn, dand any constant\(\kappa\)such that\(11\le \kappa d \le d\), ifGis a\((d,\kappa d)\)-edge colorable graphGof ordernthen there exits positive constants\(c_1,c_2\)such that for any\(\beta \le c_1 (2n)^{-c_2d^{-1}}\), any\(\beta\)-sparse list assignmentLforGis avoidable.

Corollary 2

For any positive integersn, dand any constants\(c,\kappa\)such that\(11\le \kappa d \le d\)and\(d \ge c\log n\), ifGis a\((d, \kappa d)\)-edge colorable graphGof ordernthen there exits a positive constant\(\beta\)such that any\(\beta\)-sparse list assignmentLforGis avoidable.

Note that the complete bipartite graph \(K_{d,d}\) (\(d=2^t\), \(t \in {\mathbb {N}}\)) and the d-dimensional hypercube graph \(Q_d\) (\(d \in {\mathbb {N}}\)) are both (d, d)-edge colorable graphs satisfying the condition in Corollary 2. 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 \(\beta\)-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 3

For any positive integersn, d, sand any constantcsuch that\(s\ge 11\)and\(\dfrac{1}{2} (2^{-11}c^{-1} s^2d^{-1})^{2^{-9}d^{-1}s^{2}} \ge n\), ifGis a (d, s)-edge colorable graphGof ordernthen any\(\dfrac{c}{s}\)-sparse list assignmentLforGis avoidable.

If the length of every list in L is bounded by a power of s, we have a slightly different condition as follows.

Corollary 4

For any positive integersn, d, sand any constantcsuch that\(s \ge 11\)and\(\dfrac{1}{2} (2^{-11}s^{2-c}d^{-1})^{2^{-9}d^{-1}s^{2}} \ge n\), ifGis a (d, s)-edge colorable graphGof ordernthen any\(s^{c-1}\)-sparse list assignmentLforGis avoidable.

A distance-tmatching is a matching where any two edges are at distance at least t from each other. Consider an arbitrary list assignment \(L'\) (not necessariliy \(\beta\)-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.

Corollary 5

Let\(L'\)be a list assignment for the edges of a (d, s)-edge colorable graphGsuch that for each edgeeofG, \(L'(e)\le s-1\). If every edgeesatisfying\(L'(e) \ne \emptyset\)belongs to a distance-3 matching inG, then\(L'\)is avoidable.

The results in [3] and [4] allow us to conclude that the complete bipartite graph \(K_{d,d}\) (\(d=2^t, t \in {\mathbb {N}}\)) and the hypercube graph \(Q_d\) (\(d \in {\mathbb {N}}\)) are (d, d)-edge colorable graphs. Compared to the hypercube graph \(Q_d\) (\(d \in {\mathbb {N}}\)), the complete bipartite graph \(K_{d,d}\) (\(d=2^t, t \in {\mathbb {N}}\)) is much denser, so it is interesting to see how this complete bipartite graph behaves if some edges are removed. Lemma 3 considers the case when we take a set of k standard matchings out of \(K_{d,d}\) (\(d=2^t, t \in {\mathbb {N}}\)).

Lemma 3

The graphGobtained by removingkstandard matchings from thecomplete bipartite graph\(K_{d,d}\)\((d=2^t, t \in {\mathbb {N}})\)is a\((d-k, d-k)\)-edge colorable graph.

Proof

It is straightforward that G is a \((d-k)\)-regular graph. Let h be the 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). For any 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 is \(d-1-k\), which implies that G is a \((d-k, d-k)\)-edge colorable graph. \(\square\)

Recall that the Cartesian product \(G=G_1 \square \, G_2\) of the graphs \(G_1\) and \(G_2\) is a graph whose vertex set is the Cartesian product \(V(G_1) \times V(G_2)\) 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 concerns the (d, s)-edge colorability of Cartesian products of graphs.

Lemma 4

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\(G=G_1 \square \, G_2\)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 the standard coloring of \(G_1\) and \(h_2\) be the 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 \(h(uv)=h_2(u_2v_2)\) 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 \(h(uv)=h_1(u_1v_1)\). 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 \(u_2v_2z_2t_2u_2\) 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 \(s=\min \{d_1 +s_2 ,d_2+s_1\}\). \(\square\)

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 \(a \in S\) then \(a^{-1} \in S\)). 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 \(a,b \in G\) are adjacent whenever \(\{ab^{-1}, ba^{-1}\} \subseteq S\). It is straightforward that Cay(G, S) is a d-regular graph, Lemmas 5 and 6 show that if S satisfies some further conditions then Cay(G, S) is a (d, s)-edge colorable graph.

Lemma 5

Let Cay(G, S) be an undirected Cayley graph on a groupGover the generating set\(S \subseteq G \setminus \{e\}\). If\(a=a^{-1}\)for every\(a \in S\), \(|S|=d\)and there exits a subset\(S_c \subseteq S\), \(|S_c|=s\), satisfying that every element of\(S_c\)is commutative with all elements inS, then Cay(G, S) is a (d, s)-edge colorable graph.

Proof

Let h be the proper d-edge coloring of Cay(G, S) such that every edge uv in Cay(G, S) is colored a if \(uv^{-1}=vu^{-1}=a \in S\). For an edge uv colored a, consider an arbitrary element \(b \in S_c\) and let \(z=vb\), \(t=ub\), then the edges vz and uv are colored b in h. Furthermore, since b is commutative with a, i.e. \(ab=ba\), we have \(z=vb=uab=uba=ta\). This implies that there is an edge between z and t, and this edge is colored a in h. Hence, uvztu is a 2-colored 4-cycle. Because \(|S_c|=s\), each edge of Cay(G, S) is in at least \(s-1\) 2-colored 4-cycles. It follows that Cay(G, S) is a (d, s)-edge colorable graph.

Lemma 6

Let Cay(G, S) be an undirected Cayley graph on an Abelian groupGover the generating set\(S \subseteq G \setminus \{e\}\), \(S=S^{-1}\)and\(|S|=d\). Let\(S_k=\{s_1, s_2,...,s_k\}\)be a subset ofSsuch that\(S_k \cup S^{-1}_k=S\)and\(S_k\)does not contain two different elements\(s_i \ne s_j\)satisfying that\(s_i=s^{-1}_j\). IfShas the following properties:

1. (i)

   every element\(s_i\)of\(S_k\)has even order\(d_i\)\(({s_i}^{d_i}=s_i^0=e)\);

2. (ii)

   for every element\(g \in G\), there is exactly one sequence\((x_1,x_2,\dots ,x_k)\)\((x_i \in [0,d_i -1]\)for\(i \in [1,k])\)such that\(g={s_1}^{x_1}{s_2}^{x_2}\ldots {s_k}^{x_k}\);

then Cay(G,S) is a (d, d)-edge colorable graph.

Proof

We write \(S=\{s_1,\dots ,s_k, s^{-1}_1,\dots ,s^{-1}_k\}\); 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 \(i \in [1,k]\). The condition (ii) implies that there exists exactly one sequence \((x_1,x_2,\dots ,x_k)\) such that \(u={s_1}^{x_1} \dots {s_i}^{x_i} \dots {s_k}^{x_k}\) and \(v={s_1}^{x_1}\dots {s_i}^{(x_i+1) \mod d_i}\dots {s_k}^{x_k}\). We color the edge uv by color \(s_i\) if \(x_i\) is even, and by color \(s^{-1}_i\) if \(x_i\) is odd. By repeating this for all edges of Cay(G, S), we obtain the proper d-edge coloring h.

Given an edge e of Cay(G, S), let u and v be the two endpoints of e, where \(v=us_i\) (for some \(i \in [1,k]\)). Let \(u={s_1}^{x_1}\ldots {s_i}^{x_i}\ldots {s_k}^{x_k}\); \(x_i\) is called the power of\(s_i\)in u (\(i \in [1,k]\)) and denoted by \(p_u(s_i)\). Consider an arbitrary element \(s \in S\), if \(s=s_j \in S_k\), 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 \(z=vs_j=us_is_j=us_js_i=ts_i\). 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 \(s \in S^{-1}_k\), 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. \(\square\)