Abstract
AbstractLet and let be an ‐chromatic graph with a color‐critical edge, that is, there exists such that has chromatic number . Gerbner recently conjectured that every ‐vertex maximal ‐free graph with at least edges contains an induced complete ‐partite graph on vertices. Let be a graph obtained from copies of by sharing a common edge. In this paper, we show that for all if is an ‐vertex maximal ‐free graph with at least edges, then contains an induced complete bipartite graph on vertices. We also construct graphs to show that the magnitude in is best possible, which disproves Gerbner's conjecture for .
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