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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.

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. Here 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 [

Our terminology and notation will be standard. The reader is referred to [

From the above result, we can see that the edge cover chromatic number of any graph must be equal to or, This immediately gives us a simple way of classifying graphs into two types according to. More precisely, we say that G is of CI class if, and that G is of CII class if. Wang and Liu discuss the classification problem of nearly bipartite graphs and gave some sufficient conditions for a nearly bipartite graph G to be of CI-class [

Theorem 1.1. If G is a bipartite multigraph, then G has a k-edge coloring such that the number of distinct colors represented at v is min for each.

By Theorem 1.1, for any bipartite graph G with minimum degree, it must have a -edge cover coloring. So, we can see that all bipartite graphs are of CI class. In this paper, 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.

Let be a copy of simple graph G. Let u_{i} be the vertex of G, and v_{i} be the vertex of correspond with u_{i}. A new graph, denoted by D(G) is called the double graph of G if

It is easy to see that, and we have the following result.

Theorem 2.1. Let G be a graph of CI class, then D(G) is also a graph of CI class.

Proof. It’s enough to give a -edge cover coloring of D(G) with the color set.

Since G is of CI class, G has a -edge cover coloring, and we can also give a same -edge cover coloring of with the color set. Next, we color the edges between the vertices of G and.

Let, the induced subgraph of D(G) by, denoted by. It must be a bipartite graph. By Theorem 1.1, has a -edge cover coloring with color set. Clearly, we give an edge coloring of D(G) such that each color of appears at least once at each vertex . That is, D(G) has a -edge cover coloring. This proves the Theorem.

By Theorem 2.1, for each CI class graph G, D(G) is also of CI class. Now we consider that if G is of CII class, what about D(G)? This question seems very difficult to answer in the current, but we can study some special CII class graphs. We have known that some regular graphs are of CI class and some regular graphs are of CII class [

Theorem 2.2. Let G be a k-regular graph, then the double graph D(G) is of CI class.

In order to prove Theorem 2.2, we need the following useful lemma.

Lemma 2.3. Let H be a 2k-regular graph, then H contains k edge-disjoint spanning 2-factors, and.

Proof. Let, Since H is a 2kregular graph, it must be an Euler graph. Let T be an Euler tour of H. Now, we conform a bipartite graph by T. Let,.

The vertex and are adjacent in if and only if and are adjacent sequentially in T. Clearly, must be a k-regular bipartite graph. By theorem 1.1, has k edge-disjoint perfect matches. We notice that each perfect match in is exact a spanning 2-factor of H. Then H contains k edge-disjoint spanning 2-factors, and we have. The lemma is true.

The Proof of Theorem 2.2. Since G is a k-regular graph, the double graph D(G) is a 2k-regular graph. By Lemma 2.3, D(G) contains k edge-disjoint spanning 2-factors, and . We notice that the order of D(G) is even, then each of is an even cycle. By Theorem 1.1, for, has 2-edge cover coloring with color set. So, we give a 2k-edge cover coloring of D(G) with color set.

This proves the Theorem.

It is easy to see that a polynomial time algorithm for edge cover coloring of the double graph of any regular graph can be obtained by our proof. But in general, the problem of determining the edge cover chromatic number of any graphs is NP-hard because deciding whether a 3-connected 3-regular graph G is proper 3-edge colorable is NP-complete [

Conjecture 3.1. Let G be a graph of CII class, then D(G) must be a graph of CI class.

It is seems very difficult to prove the above conjecture, but one can prove that the conjecture is true for some special CII class graphs.