Edge Colouring Reduced Indifference Graphs

Abstract

The chromatic index problem -- finding the minimum number of colours required for colouring the edges of a graph -- is still unsolved for indifference graphs, whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order. Two adjacent vertices are twins if they belong to the same maximal cliques. A graph is reduced if it contains no pair of twin vertices. We prove that every indifference graph having no twin maximum degree vertices is Class 1. We present two decomposition methods for edge colouring reduced indifference graphs. Key words. edge colourings, interval graphs, proper interval graphs, minimum proper interval graphs, reduced proper interval graphs. AMS subject classification. 05C85, 05C15, 68R10, 90C27 Submitted to WG'99 on 27 February 1999. This work was partially supported by CNPq, CAPES, FAEP, FAPESP and FAPERJ, Brazilian research agencies. y Instituto de Matem'atica and COPPE, Universidade Federal do Ri..