Vertex-critical (P_3+ℓ P_1)-free and vertex-critical (gem, co-gem)-free graphs
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A graph G is k-vertex-critical if χ(G)=k but χ(G-v)<k for all v∈ V(G) where χ(G) denotes the chromatic number of G. We show that there are only finitely many k-critical (P_3+ℓ P_1)-free graphs for all k and all ℓ. Together with previous results, the only graphs H for which it is unknown if there are an infinite number of k-vertex-critical H-free graphs is H=(P_4+ℓ P_1) for all ℓ≥ 1. We consider a restriction on the smallest open case, and show that there are only finitely many k-vertex-critical (gem, co-gem)-free graphs for all k, where gem=P_4+P_1. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of C_5. This characterization allows us to give the complete list of all k-vertex-critical (gem, co-gem)-free graphs for all k≤ 16
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