Edge and total coloring of interval graphs

An edge coloring of a graph is a function assigning colors to edges so that incident edges acquire distinct colors. The least number of colors su'cient for an edge coloring of a graph G is called its chromatic index and denoted by ′(G). Let (G) be the maximal degree of G; if ′(G) = (G), then G is said to belong to class 1, and otherwise G is said to belong to class 2. A total coloring of a graph is a function assigning colors to its vertices and edges so that adjacent or incident elements acquire distinct colors. The least number of colors su'cient for a total coloring of a graph G is called its total chromatic number and denoted by T(G). If T(G) = (G) + 1 then G is said to belong to type 1, and if (G) = (G) + 2 then G is said to belong to type 2. We consider the problem of classifying interval graphs and prove that every interval graph with odd maximal degree belongs to class 1; its edges can be colored in the minimal number of colors in time O(|VG| + |EG| + ( (G))). Then we show that the conjecture of Behzad and Vizing that T(G)6 (G) + 2 holds for interval graphs. We also prove that every interval graph with even maximal degree belongs to type and its elements can be totally colored in time O(|VG| + |EG| + ( (G))). ? 2001 Published by Elsevier Science B.V.