On orientations with forbidden out-degrees

TL;DR

This paper investigates a conjecture about orienting regular graphs to avoid certain out-degree values, extending known results to higher degrees and exploring a generalized version with variable out-degree constraints.

Contribution

The paper extends the conjecture's validity to $d leq 4$ up to $d leq 6$ and verifies the generalized version for specific graph classes and out-degree functions.

Findings

Confirmed the conjecture for $d leq 6$.

Validated the generalized version for 2-degenerate graphs.

Established cases where $F(v)$ has specific structures.

Abstract

Let $G$ be a $d$-regular graph and let $F\subseteq\{0, 1, 2, \ldots, d\}$ be a list of forbidden out-degrees. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F|<\tfrac{1}{2}d$, then $G$ should admit an $F$-avoiding orientation, i.e., an orientation where no out-degrees are in the forbidden list $F$. The conjecture is known for $d\leq 4$ due to work of Ma and Lu, and here we extend this to $d\leq 6$. The conjecture has also been studied in a generalized version, where $d, F$ are changed from constant values to functions $d(v), F(v)$ that vary over all $v\in V(G)$. We provide support for this generalized version by verifying it for some new cases, including when $G$ is 2-degenerate and when every $F(v)$ has some specific structure.