The list-coloring function of signed graphs

TL;DR

This paper extends Whitney's broken cycle theorem to signed graphs, establishing lower bounds on the number of list colorings relative to proper colorings, with improvements under specific conditions.

Contribution

It generalizes the broken cycle theorem to signed graphs and provides new bounds for list colorings depending on graph parameters and list properties.

Findings

Lower bounds on list colorings for signed graphs with specific k-values.

Extension of Whitney's broken cycle theorem to signed graphs.

Improved bounds when list assignments are 0-free or 0-included with parity conditions.

Abstract

It is known that, for any $k$-list assignment $L$ of a graph $G$, the number of $L$-list colorings of $G$ is at least the number of the proper $k$-colorings of $G$ when $k>(m-1)/\ln(1+\sqrt{2})$. In this paper, we extend the Whitney's broken cycle theorem to $L$-colorings of signed graphs, by which we show that if $k> \binom{m}{3}+\binom{m}{4}+m-1$ then, for any $k$-assignment $L$, the number of $L$-colorings of a signed graph $\Sigma$ with $m$ edges is at least the number of the proper $k$-colorings of $\Sigma$. Further, if $L$ is $0$-free (resp., $0$-included) and $k$ is even (resp., odd), then the lower bound $\binom{m}{3}+\binom{m}{4}+m-1$ for $k$ can be improved to $(m-1)/\ln(1+\sqrt{2})$.