Pseudo-multifan and Lollipop

TL;DR

This paper introduces the concepts of pseudo-multifan and lollipop as new tools for analyzing edge coloring in graphs with small core degree, aiming to address a longstanding conjecture related to overfull graphs.

Contribution

It develops the novel concepts of pseudo-multifan and lollipop and explores their properties to advance understanding of edge coloring in graphs with small core degree.

Findings

Pseudo-multifan and lollipop are effective tools in edge coloring analysis.

These concepts help characterize graphs with small core degree.

Potential progress towards the Core Conjecture in graph theory.

Abstract

A simple graph $G$ with maximum degree $\Delta$ is \emph{overfull} if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$. Clearly, the chromatic index of $G$ equals $\Delta+1$ if $G$ is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if $G$ is a simple connected graph with $\Delta\ge 3$ and $\Delta(G_\Delta)\le 2$, then $\chi'(G)=\Delta+1$ implies that $G$ is overfull or $G=P^*$, where $P^*$ 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.