Given a graph G and an edge coloring C of G, an alternating cycle of G is such a cycle of G in which any adjacent edges have distinct colors. Let dc(v), named the color degree of a vertex v, be defined as the maximum number of edges incident with v, that have distinct colors. In this paper, some color degree conditions for the existence of alternating cycles of length 3 or 4 are obtained. We al...

A graph is f -choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f . We show that the sum choice number of a 2 × n array (equivalent to list edge coloring K2,n and to list vertex coloring the cartesian product K22Kn) is n2 ...

An edge coloring of a graph G is called Mi-edge coloring if at most i colors appear at any vertex of G. Let Ki(G) denote the maximum number of colors used in an Mi-edge coloring of G. In this paper we determine the exact value of K2(G) for any graph G of maximum degree at most 3. Mathematics Subject Classification: 05C15, 05C38

An edge coloring φ of a graph G is called an Mi-edge coloring if |φ(v)| ≤ i for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let Ki(G) denote the maximum number of colors used in an Mi-edge coloring of G. In this paper we establish some bounds of K2(G), present some graphs achieving the bounds and determine exact values of K2(G) for dense graphs.

An r-edge-coloring of a graph is an assignment of r colors to the edges of the graph. An exactly r-edge-coloring of a graph is an r-edge-coloring of the graph that uses all r colors. A matching of an edge-colored graph is called rainbow matching, if no two edges have the same color in the matching. In this paper, we prove that an exactly r-edge-colored complete graph of order n has a rainbow ma...

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...