Infinite families of k-vertex-critical (P_5, C_5)-free graphs

06/06/2023

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by   Ben Cameron, et al.

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A graph is k-vertex-critical if χ(G)=k but χ(G-v)<k for all v∈ V(G). We construct a new infinite families of k-vertex-critical (P_5,C_5)-free graphs for all k≥ 6. Our construction generalizes known constructions for 4-vertex-critical P_7-free graphs and 5-vertex-critical P_5-free graphs and is in contrast to the fact that there are only finitely many 5-vertex-critical (P_5,C_5)-free graphs. In fact, our construction is actually even more well-structured, being (2P_2,K_3+P_1,C_5)-free.