Chromatic Index Critical Graphs of Odd Order with Five Major Vertices

Abstract

In an earlier paper [11], we proved that there does not exist any $\mathrm{\Delta }$-critical graph of even order with five major vertices. In this paper, we prove that if $G$ is a $\mathrm{\Delta }$-critical graph of odd order $2n+1$ with five major vertices, then $e(G)=n\mathrm{\Delta }+1$. This extends an earlier result of Chetwynd and Hilton, and also completes our characterization of graphs with five major vertices. In [9], we shall apply this result to establish some results on class 2 graphs whose core has maximum degree two.