Alternating odd cycles and orientations of Kneser-like graphs

TL;DR

This paper explores orientations of topologically χ-chromatic graphs, like Kneser and Schrijver graphs, to avoid or control alternating odd cycles, revealing new structural insights and coloring bounds.

Contribution

It introduces the concept of alternating odd cycles in orientations and investigates their presence in topologically χ-chromatic graphs, providing new results for several graph families.

Findings

Certain orientations avoid alternating odd cycles in specific graph classes

All shortest odd cycles can be made alternating in some graphs

Results establish new bounds on graph coloring using topological methods

Abstract

We call an oriented odd cycle alternating if it has exactly one vertex whose in-degree and out-degree are both positive. In this paper, we investigate whether certain graphs admit an orientation that avoids alternating odd cycles as subgraphs, or one in which all their shortest odd cycles become alternating. Our focus is on topologically $\chi$-chromatic graphs, that is, graphs for which the topological method yields a sharp lower bound on the chromatic number. We present results for several graph families, including Kneser graphs, Schrijver graphs, and generalized Mycielski graphs.