Counting odd cycle free orientations of graphs

TL;DR

This paper determines the maximum number of orientations of an n-vertex graph that contain no strongly connected odd cycles, addressing a recent open question in graph orientation theory.

Contribution

It provides a precise maximum count for orientations avoiding strongly connected odd cycles, advancing understanding of graph orientations and cycle constraints.

Findings

Identifies the maximum number of such orientations for any n-vertex graph.

Answers a specific open question in the study of graph orientations.

Contributes to the theory of cycle-avoiding orientations in graphs.

Abstract

In this short note we determine the maximum number, over all $n$-vertex graphs $G$, of orientations of $G$ containing no strongly connected cycle $C_{2k+1}$. This answers a part of a recent question of Araujo, Botler and Mota.