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. We present new positive evidence for the conjecture: every non neighbourhood-overfull indifference graph can be edge coloured with maximum degree colours. 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. A graph is overfull if the total number of edges is greater than the product of the maximum degree by ⌊n/2⌋, where n is the number of vertices. We give a structural characterization for neighbourhood-overfull indifference graphs proving that a reduced indifference graph cannot be neighbourhood-overfull. We show that the chromatic index for all reduced indifference graphs is the maximum degree. We present two decomposition methods for edge colouring reduced indifference graphs with maximum degree colours. © 2002 Elsevier Science B.V. All rights reserved.29701/03/15145155Cai, L., Ellis, J.A., NP-completeness of edge-colouring some restricted graphs (1991) Discrete Appl. Math., 30, pp. 15-27De Figueiredo, C.M.H., Meidanis, J., De Mello, C.P., A linear-time algorithm for proper interval graph recognition (1995) Inform. Process. Lett., 56, pp. 179-184De Figueiredo, C.M.H., Meidanis, J., De Mello, C.P., On edge-colouring indifference graphs (1997) Theoret. Comput. Sci., 181, pp. 91-106De Figueiredo, C.M.H., Meidanis, J., De Mello, C.P., Total-chromatic number and chromatic index of dually chordal graphs (1999) Inform. Process. Lett., 70, pp. 147-152De Figueiredo, C.M.H., Meidanis, J., De Mello, C.P., Local conditions for edge-coloring (2000) J. Combin. Math. Combin. Comput., 32, pp. 79-91. , Tech. Rep., DCC 17/95, UNICAMP, 1995Gutierrez, M., Oubiña, L., Minimum proper interval graphs (1995) Discrete Math., 142, pp. 77-85Hammer, P.L., Peled, U.N., Sun, X., Difference graphs (1990) Discrete Appl. Math., 28, pp. 35-44Hedman, B., Clique graphs of time graphs (1984) J. Combin. Theory Ser. B, 37, pp. 270-278Hilton, A.J.W., Two conjectures on edge-colouring (1989) Discrete Math., 74, pp. 61-64Holyer, I., The NP-completeness of edge-coloring (1981) SIAM J. Comput., 10, pp. 718-720Misra, J., Gries, D., A constructive proof of Vizing's theorem (1992) Inform. Process. Lett., 41, pp. 131-133Roberts, F.S., On the compatibility between a graph and a simple order (1971) J. Combin. Theory Ser. B, 11, pp. 28-38Vizing, V.G., On an estimate of the chromatic class of a p-graph (1964) Diskrete Anal., 3, pp. 25-30. , (in Russian)