Kempe Equivalent List Edge-Colorings of Planar Graphs

TL;DR

This paper proves that line graphs of large-degree planar graphs are highly flexible in list colorings, allowing any two valid colorings to be transformed into each other through Kempe swaps under certain list assignment conditions.

Contribution

It establishes new conditions under which line graphs of planar graphs are $L$-swappable, extending known results in list coloring and Kempe equivalence for such graphs.

Findings

Line graphs of planar graphs with max degree ≥ 9 are $L$-swappable with $( ext{max degree}+1)$-list assignments.

Line graphs of planar graphs with max degree ≥ 15 are $L$-swappable with $ ext{max degree}$-list assignments.

Results are analogous to and extend previous $L$-choosability theorems by Borodin and others.

Abstract

For a list assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap in $\varphi$ is \emph{$L$-valid} if it yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from another by a sequence of $L$-valid Kempe swaps. And a graph $G$ is \emph{$L$-swappable} if every two of its $L$-colorings are $L$-equivalent. We consider $L$-swappability of line graphs of planar graphs with large maximum degree. Let $G$ be a planar graph with $\Delta(G)\ge 9$ and let $H$ be the line graph of $G$. If $L$ is a $(\Delta(G)+1)$-assignment to $H$, then $H$ is $L$-swappable. Let $G$ be a planar graph with $\Delta(G)\ge 15$ and let $H$ be the line graph of $G$. If $L$ is a $\Delta(G)$-assignment to $H$, then $H$ is $L$-swappable. The first result is analogous to one for $L$-choosability by Borodin, which was later strengthened by Bonamy. The second result is analogous to another for $L$-choosability by Borodin, which was later strengthened by Borodin, Kostochka, and Woodall.