arXiv subjects
Publications and source records attributed to Joshua Harrelson.
For a graph $G$, we show that if $mad(G)<m$, then $\chi'_\ell(G)\leq \Delta+1$ where $m$ depends upon $\Delta$ and $\chi'_\ell(G)$ is the list-chromatic index of $G$. When $\Delta\leq 20$ the value of $m$ is close to $\frac{1}{2}\Delta$, but as $\Delta$ increases $m$ becomes asymptotic to about $\frac{1}{4}\Delta+5$.
We prove that triangulations with maximum degree at most 5 satisfy the List-Edge-Coloring Conjecture.
Let $G$ be a simple planar graph of maximum degree $Δ$, let $t$ be a positive integer, and let $L$ be an edge list assignment on $G$ with $|L(e)| \geq Δ+t$ for all $e \in E(G)$. We prove that if $H$ is a subgraph of $G$ that has been $L$-edge-coloured, then the edge-precolouring can be extended to an $L$-edge-colouring of $G$, provided that $H$ has maximum degree $d\leq t$ and either $d \leq t-4$ or $Δ$ is large enough ($Δ\geq 16+d$ suffices). If $d>t$, there are examples for any choice of $Δ$ where the extension is impossible.