An Alon-Tarsi Style Theorem for Additive Colorings

TL;DR

This paper provides an alternative proof of the Alon-Tarsi list coloring theorem and extends it to additive colorings using a new digraph construction, leading to new list coloring results for specific graph classes.

Contribution

It introduces an additive coloring analog of the Alon-Tarsi theorem by constructing a new digraph and analyzing Eulerian subdigraphs, expanding the theorem's applicability.

Findings

Established an additive coloring version of the Alon-Tarsi theorem.

Proved additive list coloring results for certain tripartite graphs.

Developed a new digraph construction for analyzing additive colorings.

Abstract

We first give an alternative proof of the Alon-Tarsi list coloring theorem. We use the ideas from this proof to obtain the following result, which is an additive coloring analog of the Alon-Tarsi Theorem: Let $G$ be a graph and let $D$ be an orientation of $G$. We introduce a new digraph $\mathcal{W}(D)$, such that if the out-degree in $D$ of each vertex $v$ is $d_v$, and if the number of Eulerian subdigraphs of $\mathcal{W}(D)$ with an even number of edges differs from the number of Eulerian subdigraphs of $\mathcal{W}(D)$ with an odd number of edges, then for any assignment of lists $L(v)$ of $d_v+1$ positive integers to the vertices of $G$, there is an additive coloring of $G$ assigning to each vertex $v$ an element from $L(v)$. As an application, we prove an additive list coloring result for tripartite graphs $G$ such that one of the color classes of $G$ contains only vertices whose neighborhoods are complete.