Strong edge-colorings for k-degenerate graphs

We prove that the strong chromatic index for each k-degenerate graph with maximum degree ∆ is at most (4k − 2)∆ − k(2k − 1) + 1. A strong edge-coloring of a graph G is an edge-coloring so that no edge can be adjacent to two edges with the same color. So in a strong edge-coloring, every color class gives an induced matching. The strong chromatic index χs(G) is the minimum number of colors needed to color E(G) strongly. This notion was introduced by Fouquet and Jolivet (1983, [5]). Erdős and Nešetřil during a seminar in Prague in 1985 proposed some open problems, one of which is the following Conjecture 1 (Erdős and Nešetřil, 1985). If G is a simple graph with maximum degree ∆, then χs(G) ≤ 5∆/4 if ∆ is even, and χs(G) ≤ (5∆ − 2∆ + 1)/4 if ∆ is odd. This conjecture is true for ∆ ≤ 3 ([1, 6]). Cranston [4] showed that χs(G) ≤ 22 for ∆ = 4. Chung, Gyárfás, Trotter, and Tuza (1990, [3]) showed that the upper bounds are exactly the numbers of edges in 2K2-free graphs. Molloy and Reed [8] proved that graphs with sufficient large maximum degree ∆ has strong chromatic index at most 1.998∆. For more results see [9] (Chapter 6, problem 17). A graph is k-degenerate if every subgraph has minimum degree at most k. Chang and Narayanan (2012, [2]) recently proved that a 2-degenerate graph with maximum degree ∆ has strong chromatic index at most 10∆− 10. Luo and the author in [7] improved the upper bound to 8∆− 4. In [2], the following conjecture was made Conjecture 2 (Chang and Narayanan, [2]). There exists an absolute constant c such that for any kdegenerate graphs G with maximum degree ∆, χs(G) ≤ ck∆. Furthermore, the k may be replaced by k. In this paper, we prove a stronger form of the conjecture. Unlike the priming processes in[2, 7], we find a special ordering of the edges and by using a greedy coloring obtain the following result. Theorem 1. The strong chromatic index for each k-degenerate graph with maximum degree ∆ is at most (4k − 2)∆− k(2k − 1) + 1. Thus, 2-degenerate graphs have strong chromatic index at most 6∆− 5. Proof. By definition of k-degenerate graphs, after the removal of all vertices of degree at most k, the remaining graph has no edges or has new vertices of degree at most k, thus we have the following simple fact on kdegenerate graphs (see also [2]). Let G be a k-degenerate graph. Then there exists u ∈ V (G) so that u is adjacent to at most k vertices of degree more than k. Moreover, if ∆(G) > k, then the vertex u can be selected with degree more than k. We call a vertex u a special vertex if u is adjacent to at most k vertices of degree more than k. An edge is a special edge if it is incident to a special vertex and a vertex with degree at most k. The above fact implies that every k-degenerate graph has a special edge, and if ∆ ≤ k, then every vertex and every edge are special. We order the edges of G as follows. First we find in G a special edge, put it at the beginning of the list, and then remove it from G. Repeat the above step in the remaining graph. When the process ends, we have an ordered list of the edges in G, say e1, e2, . . . , em, where m = |E(G)|. So em is the special edge we first chose and placed in the list. Date: March 29, 2013. The author’s research was supported in part by an NSA grant.