List-avoiding orientations

TL;DR

This paper advances the understanding of $F$-avoiding orientations in graphs by proving new bounds on forbidden value sets relative to vertex degrees, using algebraic combinatorics tools like the Combinatorial Nullstellensatz.

Contribution

It improves existing bounds for the existence of $F$-avoiding orientations, moving closer to a conjecture that relates forbidden set sizes to vertex degrees.

Findings

Proved that if $|F(v)| < loor{rac{1}{3} deg(v)}$, then an $F$-avoiding orientation exists.

Showed that under certain degree conditions, the coefficient can be increased to approximately 0.414.

Developed a new sufficient condition based on the Combinatorial Nullstellensatz for such orientations.

Abstract

Given a graph $G$ with a set $F(v)$ of forbidden values at each $v \in V(G)$, an $F$-avoiding orientation of $G$ is an orientation in which $deg^+(v) \not \in F(v)$ for each vertex $v$. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F(v)| < \frac{1}{2} deg(v)$ for each $v \in V(G)$, then $G$ has an $F$-avoiding orientation, and they showed that this statement is true when $\frac{1}{2}$ is replaced by $\frac{1}{4}$. In this paper, we take a step toward this conjecture by proving that if $|F(v)| < \lfloor \frac{1}{3} deg(v) \rfloor$ for each vertex $v$, then $G$ has an $F$-avoiding orientation. Furthermore, we show that if the maximum degree of $G$ is subexponential in terms of the minimum degree, then this coefficient of $\frac{1}{3}$ can be increased to $\sqrt{2} - 1 - o(1) \approx 0.414$. Our main tool is a new sufficient condition for the existence of an $F$-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi.