On the deficiency of complete multipartite graphs
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An edge-coloring of a graph G with colors 1,...,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 integer interval. It is well-known that there are graphs that do not have interval colorings. The deficiency of a graph G, denoted by def(G), is the minimum number of pendant edges whose attachment to G leads to a graph admitting an interval coloring. In this paper we investigate the problem of determining or bounding of the deficiency of complete multipartite graphs. In particular, we obtain a tight upper bound for the deficiency of complete multipartite graphs. We also determine or bound the deficiency for some classes of complete multipartite graphs.
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