Chromatic-index-critical graphs of orders 13 and 14

Abstract

A graph is chromatic-index-critical if it cannot be edge-coloured with Δ colours (with Δ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order [ [18] A.J.W. Hilton, R.J. Wilson, Edge-colorings of graphs: a progress report, in: M.F. Cabobianco, et al. (Eds.), Graph Theory and its Applications: East and West, New York, 1989, pp. 241–249; [31] H.P. Yap, Some topics in graph theory, London Mathematical Society, Lecture Note Series, vol. 108, Cambridge University Press, Cambridge, 1986]. In this paper we show that there are no chromatic-index-critical graphs of order 14. Our result extends that of [ [5] G. Brinkmann, E. Steffen, Chromatic-index-critical graphs of orders 11 and 12, European J. Combin. 19 (1998) 889–900] and leaves order 16 as the only case to be checked in order to decide on the minimality of the counterexamples given by Chetwynd and Fiol. In addition we list all nontrivial critical graphs of order 13.