»ã±¨±êÌâ (Title)£º½ûÓÃһЩÆæȦͼµÄ×îÓ׶Ȳ»±äÐÔ(Minimum degree stability of graphs forbidding some odd cycles)

»ã±¨ÈË (Speaker)£ºÅíÔÀ½¨ ½ÌÊÚ£¨ºþÄÏ´óѧ£©

»ã±¨¹¦·ò (Time)£º2023Äê4ÔÂ19ÈÕ(ÖÜÈý) 19:30

»ã±¨µØÖ· (Place)£ºÌÚѶ»áÒé 739-770-096

Ô¼ÇëÈË(Inviter)£º¿Â·öÓ¢ ½ÌÊÚ

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»ã±¨ÌáÒª£º We 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 $C_{2k+1}$-free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andr\'{a}sfai, Erd\H{o}s and S\'{o}s showed that if a $\{C_3,C_5,\cdots, C_{2k+1}\}$-free graph on $n$ vertices has minimum degree greater than $\frac{2}{2k+3}n$, then it is bipartite. H\"{a}ggkvist showed that for $k\in \{1,2,3,4\}$, if a $C_{2k+1}$-free graph on $n$ vertices has minimum degree greater than $\frac{2}{2k+3}n$, then it is bipartite. H\"{a}ggkvist also pointed out that this result cannot be extended to $k\geq 5$. In this paper, we give a complete answer for any $k\geq 5$.