Let G be a (simple) graph with maximum degree three and chromatic index four. A 3-edge-coloring of G is a coloring of its edges in which only three colors are used. Then a vertex is conflicting when some edges incident to it have the same color. The minimum possible number of conflicting vertices that a 3edge-coloring of G can have, d(G), is called the edge-coloring degree of G. Here we are mai...

Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω( f ) of an edge-coloring f of G is the sum of costs ω( f (e)) of colors f (e) assigned to all edges e in G. An edge-coloring f of G is optimal if ω( f ) is minimum amon...

In a proper edge-coloring of cubic graph, an edge e is normal if the set colors used by five edges incident with end has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless graph 5-edge-coloring, is, 5-edge-coloring such all are normal. this paper, we prove result related to conjecture. parameter μ3 measurement for graphs, introduced Steffen in 2015. Our shows G a...

A mixed hypergraph H = (X, C,D) consists of the vertex set X and two families of subsets: the family C of C-edges and the family D of D-edges. In a coloring, every C-edge has at least two vertices of common color, while every D-edge has at least two vertices of different colors. The largest (smallest) number of colors for which a coloring of a mixed hypergraph H using all the colors exists is c...

Kostochka, A.V., List edge chromatic number of graphs with large girth, Discrete Mathematics 101 (1992) 189-201. It is shown that the list edge chromatic number of any graph with maximal degree A and girth at least 8A(ln A + 1.1) is equal to A + 1 or to A.

We consider the following type of question: Given a partial proper d-edge coloring of the d-dimensional hypercube Qd, and lists of allowed colors for the non-colored edges of Qd, can we extend the partial coloring to a proper d-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy ...

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The chromatic edge strength of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with paralle...