On Deeply Critical Oriented Cliques

TL;DR

This paper investigates deeply critical oriented graphs, proving their existence for all odd orders greater than or equal to 9 and their non-existence among certain even-order circulant oriented cliques, thus resolving an open problem.

Contribution

It establishes the existence of deeply critical oriented cliques for all odd orders ≥ 9 and shows they do not exist among even-order circulant oriented cliques, answering a longstanding question.

Findings

Deeply critical oriented cliques exist for all odd n ≥ 9.

No deeply critical circulant oriented cliques of even order exist.

The results close an open problem in the theory of oriented graph colorings.

Abstract

In this work we consider arc criticality in colourings of oriented graphs. We study deeply critical oriented graphs, those graphs for which the removal of any arc results in a decrease of the oriented chromatic number by $2$. We prove the existence of deeply critical oriented cliques of every odd order $n\geq 9$, closing an open question posed by Borodin et al. (Journal of Combinatorial Theory, Series B, 81(1):150-155, 2001). Additionally, we prove the non-existence of deeply critical oriented cliques among the family of circulant oriented cliques of even order.