Edge and total coloring of interval graphs

Abstract

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 sufficient for an edge coloring of a graph G is called its chromatic index and denoted by χ′( G). Let Δ( G) be the maximal degree of G; if χ′( G)= Δ( 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 sufficient for a total coloring of a graph G is called its total chromatic number and denoted by χ T( G). If χ T( G)= Δ( G)+1 then G is said to belong to type 1, and if χ( G)= Δ( 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(| V G |+| E G |+( Δ( G)) 2). Then we show that the conjecture of Behzad and Vizing that χ T( G)⩽ Δ( 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(| V G |+| E G |+( Δ( G)) 2).