Pseudo-multifan and Lollipop

Abstract

A simple graph $G$ with maximum degree $\hat{I}”$ is \emph{overfull} if $|E(G)|>\hat{I}”\lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_{\hat{I}}”$, is the subgraph of $G$ induced by its vertices of degree $\hat{I}”$. Clearly, the chromatic index of $G$ equals $\hat{I}”+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $\hat{I}”\ge 3$ and $\hat{I}”(G_{\hat{I}}”)\le 2$, then $\ddot{I}{‡}^{\prime }(G)=\hat{I}”+1$ implies that $G$ is overfull or $G=P^{\ast }$, where $P^{\ast }$ is obtained from the Petersen graph by deleting a vertex (Core Conjecture). The goal of this paper is to develop the concepts of ``pseudo-multifan'' and ``lollipop'' and study their properties in an edge colored graph. These concepts turn out to be powerful tools in edge coloring graphs with a small core degree.

This is the first split of arXiv:2004.00734