The Core Conjecture of Hilton and Zhao I: Pseudo-multifan and Lollipop

A simple graph $G$ with maximum degree $Delta$ is overfull if $|E(G)|>Delta lfloor |V(G)|/2rfloor$. The 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 $Deltage 3$ 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 and properties are of independent interests, and will be particularly used to prove the Core Conjecture in a subsequent paper.