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arXiv · 2102.03855

The stability method, eigenvalues and cycles of consecutive lengths

Binlong Li·Bo Ning
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Abstract

Woodall proved that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is an integer, if $e(G)\geq \binom{n-k-1}{2}+\binom{k+2}{2}+1$ then $G$ contains a $C_{\ell}$ for each $\ell\in [3,n-k]$. In this article, we prove a stability result of this theorem. As a byproduct, we give complete solutions to two problems in \cite{GN19}. Our second part is devoted to an open problem by Nikiforov: what is the maximum $C$ such that for all positive $\varepsilon \sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$ for every $\ell\leq (C-\varepsilon)n$. We prove that $C\geq\frac{1}{4}$ by a method different from previous ones, improving the existing bounds. We also derive an Erd\H{o}s-Gallai type edge number condition for even cycles, which may be of independent interest.

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Binlong Li, Bo Ning. 2021-02-07. The stability method, eigenvalues and cycles of consecutive lengths. https://arxiv.org/abs/2102.03855

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