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A simple graph $G$ with maximum degree $\Delta$ is overfull if $|E(G)|>\Delta\lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_{\Delta}$, is thesubgraph of $G$ induced by its vertices of degree $\Delta$. Clearly, thechromatic index of $G$ equals $\Delta+1$ if $G$ is overfull. Conversely, Hiltonand Zhao in 1996 conjectured that if $G$ is a simple connected graph with$\Delta\ge 3$ and $\Delta(G_\Delta)\le 2$, then $\chi'(G)=\Delta+1$ impliesthat $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersengraph by deleting a vertex (Core Conjecture). The goal of this paper is todevelop the concepts of "pseudo-multifan" and "lollipop" and study theirproperties in an edge colored graph. These concepts and properties are ofindependent interests, and will be particularly used to prove the CoreConjecture in a subsequent paper.

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