A refinement on the structure of vertex-critical (P_5, gem)-free graphs
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We give a new, stronger proof that there are only finitely many k-vertex-critical (P_5, gem)-free graphs for all k. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all 6- and 7-vertex-critical (P_5, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the k-colourability of (P_5, gem)-free graphs for all k where the certificate is either a k-colouring or a (k+1)-vertex-critical induced subgraph. Our complete lists for k≤ 7 allow for the implementation of these algorithms for all k≤ 6.
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