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- W3127915511 abstract "Woodall proved that for a graph $G$ of order $ngeq 2k+3$ where $kgeq 0$ 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 $ellin [3,n-k]$. 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 $varepsilon sqrt{lfloorfrac{n^2}{4}rfloor}$ contains a cycle of length $ell$ for every $ellleq (C-varepsilon)n$. We prove that $Cgeqfrac{1}{4}$ by a method different from previous ones, improving the existing bounds. We also derive an ErdH{o}s-Gallai type edge number condition for even cycles, which may be of independent interest." @default.
- W3127915511 created "2021-02-15" @default.
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- W3127915511 date "2021-02-07" @default.
- W3127915511 modified "2023-09-27" @default.
- W3127915511 title "The stability method, eigenvalues and cycles of consecutive lengths" @default.
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