  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.
