Some Remarks on a Paper of Chetwynd and Hilton on Critical Star Multigraphs

Abstract

In [A.G. Chetwynd and A.J.W. Hilton, Critical star multigraphs, Graphs and Combinatorics 2(1986), 209-221], Chetwynd and Hilton started the investigations of the edge-chromatic properties of a particular class of multigraphs, which they called star multigraphs. A star multigraph is a multigraph such that there exists a vertex $v^{\ast }$ that is incident with each multiple edge. Star multigraphs turn out to be useful tools in the study of the chromatic index of simple graphs.

The main goal of this paper is to provide shorter and simpler proofs of all the main theorems contained in the above-mentioned paper. Most simplifications are achieved by means of a formula for the chromatic index recently obtained by the author and by a careful use of arguments involving fans.