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
≥
9, closing an open question posed by Borodin et al. Additionally, we prove the nonexistence of deeply critical oriented cliques among the family of circulant oriented cliques of even order.
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