We present a deterministic distributed algorithm, in the LOCALmodel, that computes a (1+o(1))∆edge-coloring in polylogarithmic-time, so long as the maximum degree ∆ = Ω̃(logn). For smaller ∆, we give a polylogarithmic-time 3∆/2-edge-coloring. These are the first deterministic algorithms to go below the natural barrier of 2∆− 1 colors, and they improve significantly on the recent polylogarithmict...

A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index of a graph is the minimum number of colors needed to obtain a strong edge coloring. In an analogous way, we can define the list version of strong edge coloring and list version of strong chromatic index. In this paper we prove that if G is a graph with maximu...

Giving a planar graph G, let χl(G) and χ ′′ l (G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if a planar graph G without 6-cycles with chord, then χl(G) ≤ ∆(G) + 1 and χ ′′ l (G) ≤ ∆(G) + 2 where ∆(G) ≥ 6.

An edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if G has an interval t-coloring for some positive integer t. Let N be the set of all interval colorable graphs. For a graph G ∈ N, the least and the greatest va...

A proper edge-coloring of a graph G with colors 1, . . . , t is called an interval t-coloring if the colors of edges incident to any vertex of G form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. For an interval colorable graph G, the least value of t for which G has an interval t-coloring is denoted by w(G). A graph G is ...

A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that is adjacent to a vertex of every other color. The b-chromatic number b(G) of a graph G is the largest k such that G admits a b-coloring with k colors. A graph G is b-chromatic edge critical if for any edge e of G, the b-chromatic number of G − e is less than the b-chromatic number of G. We ...

A decomposition of λ copies of monochromatic Kv into copies of K4 such that each copy of K4 contains at most one edge from each Kv is called a proper edge coloring of a BIBD(v, 4, λ). We show that the necessary conditions are sufficient for the existence of a BIBD(v, 4, λ) which has such a proper edge coloring.

Galvin ([7]) proved that every k-edge-colorable bipartite multigraph is kedge-choosable. Slivnik ([11]) gave a streamlined proof of Galvin's result. A multigraph G is said to be nearly bipartite if it contains a special vertex Vs such that G Vs is a bipartite multigraph. We use the technique in Slivnik's proof to obtain a list coloring analog of Vizing's theorem ([12]) for nearly bipartite mult...

Graph coloring is a central problem in distributed computing. Both vertexand edge-coloring problems have been extensively studied in this context. In this paper we show that a (2∆ − 1)-edge-coloring can be computed in time smaller than log n for any > 0, specifically, in e √ log logn) rounds. This establishes a separation between the (2∆ − 1)-edge-coloring and Maximal Matching problems, as the ...

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let Q be an additive hereditary property of graphs. A Q-edge-coloring of a simple graph is an edge coloring in which the edges colored with the same color induce a subgraph of property Q. In this paper we present some results on fractional Q-edge-colorings. We determine...