Abstract
AbstractWe consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a ‐free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist showed that for , if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist also pointed out that this result cannot be extended to . In this paper, we give a complete answer for any . We show that if and is an ‐vertex ‐free graph with , then is bipartite, and the bound is tight.
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