On Deeply Critical Oriented Cliques

03/31/2021

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by   Christopher Duffy, et al.

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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≥ 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.