Given graphs G and H, we consider the problem of decomposing a properly edge-colored graph G into few parts consisting of rainbow copies of H and single edges. We establish a close relation to the previously studied problem of minimum H-decompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of H and single edges.

We study weighted bipartite edge coloring problem, which is a generalization of two classical problems: bin packing and edge coloring. This problem has been inspired from the study of Clos networks in multirate switching environment in communication networks. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite multigraph G = (V,E) with weights w : E → [0, 1]. Th...

An edge-coloring of a graph G with consecutive integers c1, . . . , ct is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. The set of all interval colorable graphs is denoted by N. In 2004, Giaro and...

An edge-coloring of a graph G with colors 1, 2, . . . , 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 it has an interval t-coloring for some positive integer t. For an interval colorable graph G, W (G) denotes the greatest value of t for which G has an int...

Abs t r ac t . In an edge-coloring of a graph G = (V, E) each color appears around each vertex at most once. An f-coloring is a generalization of an edge-coloring in which each color appears around each vertex v at most f(v) times where f is a function assigning a natural number f(v) e N to each vertex v E V. In this paper we first give a simple reduction of the f-coloring problem to the ordina...

Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic...

A k-edge-coloring of a graph G = (V, E) is a function c that assigns an integer c(e) (called color) in {0, 1, · · · , k−1} to every edge e ∈ E so that adjacent edges get different colors. A k-edge-coloring is linear compact if the colors incident to every vertex are consecutive. The problem k − LCCP is to determine whether a given graph admits a linear compact k-edge coloring. A k-edge-coloring...

If c : E → {1, 2, . . . , k} is a proper edge coloring of a graph G = (V,E) then the palette S(v) of a vertex v ∈ V is the set of colors of the incident edges: S(v) = {c(e) : e = vw ∈ E}. An edge coloring c distinguishes vertices u and v if S(u) 6= S(v). A d-strong edge coloring of G is a proper edge coloring that distinguishes all pairs of vertices u and v with distance d(u, v) ≤ d. The minimu...

We prove that the list chromatic index of a graph of maximum degree and treewidth ≤ √2 − 3 is ; and that the total chromatic number of a graph of maximum degree and treewidth ≤ /3 + 1 is + 1. This improves results by Meeks and Scott. C © 2015 Wiley Periodicals, Inc. J. Graph Theory 81: 272–282, 2016

We prove that the list chromatic index of a graph of maximum degree ∆ and treewidth ≤ √ 2∆ − 3 is ∆; and that the total chromatic number of a graph of maximum degree ∆ and treewidth ≤ ∆/3 + 1 is ∆ + 1. This improves results by Meeks and Scott.