1. Minimum degree stability of C2k+1 ${C}_{2k+1}$‐free graphs.
- Author
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Yuan, Xiaoli and Peng, Yuejian
- Abstract
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 C2k+1 ${C}_{2k+1}$‐free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a {C3,C5,...,C2k+1} $\{{C}_{3},{C}_{5},\ldots ,{C}_{2k+1}\}$‐free graph on n $n$ vertices has minimum degree greater than 22k+3n $\frac{2}{2k+3}n$, then it is bipartite. Häggkvist showed that for k∈{1,2,3,4} $k\in \{1,2,3,4\}$, if a C2k+1 ${C}_{2k+1}$‐free graph on n $n$ vertices has minimum degree greater than 22k+3n $\frac{2}{2k+3}n$, then it is bipartite. Häggkvist also pointed out that this result cannot be extended to k≥5 $k\ge 5$. In this paper, we give a complete answer for any k≥5 $k\ge 5$. We show that if k≥5 $k\ge 5$ and G $G$ is an n $n$‐vertex C2k+1 ${C}_{2k+1}$‐free graph with δ(G)≥n6+1 $\delta (G)\ge \frac{n}{6}+1$, then G $G$ is bipartite, and the bound n6+1 $\frac{n}{6}+1$ is tight. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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