Abstract
Let G be a simple graph with vertex set V(G) and edge set E(G). An edge coloring C of G is called an edge cover coloring, if each color appears at least once at each vertex . The maximum positive integer k such that G has a k edge cover coloring is called the edge cover chromatic number of G and is denoted by . It is known that for any graph G, . If , then G is called a graph of CI class, otherwise G is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification on double graph of some graphs and a polynomial time algorithm can be obtained for actually finding such a classification by our proof.
Highlights
The edge coloring problem finds a partition of all the edges in a graph into a collection of subsets of edges such that, for each subset in the partition, no edges share a common vertex
We discuss the classification problem on double graph of some graphs, and a good algorithm for edge cover coloring on double graph of k-regular graph can be obtained by the proof of theorem
We have known that some regular graphs are of CI class and some regular graphs are of CII class [7]
Summary
The edge coloring problem finds a partition of all the edges in a graph into a collection of subsets of edges such that, for each subset in the partition, no edges share a common vertex. The objective is to minimize the number of subsets in a partition. This problem has interesting real life applications in the optimization and the network design, such as the file transfers in computer networks [1]. The edge coloring of graphs in this paper are not necessarily proper. By Theorem 1.1, for any bipartite graph G with minimum degree , it must have a -edge cover coloring. We discuss the classification problem on double graph of some graphs, and a good algorithm for edge cover coloring on double graph of k-regular graph can be obtained by the proof of theorem
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
