The existence of $\{p,q\}$-orientations in edge-connected graphs

TL;DR

This paper establishes a new sufficient edge-connectivity condition for the existence of specific orientations in graphs, where each vertex's out-degree is restricted to two values, generalizing previous results on modulo orientations.

Contribution

It provides a novel sufficient condition for {p,q}-orientations in edge-connected graphs, extending Thomassen's 2012 theorem on modulo orientations.

Findings

Provides a sufficient edge-connectivity criterion for {p,q}-orientations.

Generalizes Thomassen's theorem to broader conditions.

Establishes existence conditions based on vertex degree bounds and total edge count.

Abstract

In 1976 Frank and Gy{\'a}rf{\'a}s gave a necessary and sufficient condition for the existence of an orientation in an arbitrary graph $G$ such that for each vertex $v$, the out-degree $d^+_G(v)$ of it satisfies $p(v)\le d^+_G(v)\le q(v)$, where $p$ and $q$ are two integer-valued functions on $V(G)$ with $p\le q$. In this paper, we give a sufficient edge-connectivity condition for the existence of an orientation in $G$ such that for each vertex $v$, $d^+_G(v)\in \{p(v),q(v)\}$, provided that for each vertex $v$, $p(v)\le \frac{1}{2}d_G(v) \le q(v)$, $|q(v)-p(v)|\le k$, and there is $t(v)\in \{p(v),q(v)\}$ in which $|E(G)|=\sum_{v\in V(G)}t(v)$. This result is a generalization of a theorem due to Thomassen (2012) on the existence of modulo orientations in highly edge-connected graphs.