Consecutive colorings of the edges of general graphs

Given an n-vertex graph G, an edge-coloring of G with natural numbers is a consecutive (or interval) coloring if the colors of edges incident with each vertex are distinct and form an interval of integers. In this paper we prove that if G has a consecutive coloring and n¿3 then S(G)62n − 4, where S(G) is the maximum number of colors allowing a consecutive coloring. Next, we investigate the so-called de*ciency of G, a natural measure of how far it falls of being consecutively colorable. Informally, we de*ne the de*ciency def (G) of G as the minimum number of pendant edges which would need to be attached in order that the resulting supergraph has such a coloring, and compute this number in the case of cycles, wheels and complete graphs. c © 2001 Elsevier Science B.V. All rights reserved.