Pseudo-multifan and Lollipop

Abstract

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

Figures (2)

• Figure 1: (a) A typical multifan $F_\varphi(r, s_1:s_\alpha:s_\beta)$, where $\overline{\varphi}(r)=1$ and $\overline{\varphi}(s_1)=\{2,\Delta\}$; (b) A lollipop centered at $r$, where $x$ can be the same as some $s_\ell$ for $\ell\in [\beta+1, \Delta-2]$; (c) A rotation centered at $r$, where a dashed line at a vertex indicates a color missing at the vertex.
• Figure 2: Coloring operations in the proof of Lemma \ref{['Lemma:extended multifan']} (b).

Theorems & Definitions (21)

• Conjecture 1.1: Core Conjecture
• Definition 2.1: Multifan
• Definition 2.2: Pseudo-multifan
• Lemma 2.3
• Definition 2.4: Lollipop
• Theorem 2.5
• Theorem 2.6
• Corollary 2.7
• Theorem 2.8
• Lemma 3.1: StiebSTF-Book