The maximum number of cliques in graphs with bounded odd circumference

Lv, Zequn; Győri, Ervin; He, Zhen; Salia, Nika; Xiao, Chuanqi; Xiutao, Zhu,

doi:10.48550/arxiv.2212.01989

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“…After the first version of this paper was written and submitted for publication, we have learned our paper was cited in [16], and the method developed here (together with some powerful technique developed by Nikiforov in [9]) has been used by Zhang and Zhao [16] to obtain a spectral condition for the existence of cycles with consecutive odd lengths in non-bipartite graphs. Furthemore, our Lemma 2 was generalized in a recent paper [8] on generalized Turan-type problems on cycles.…”

Section: Note Addedmentioning

confidence: 99%

Eigenvalues and cycles of consecutive lengths

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2023

Journal of Graph Theory

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As the counterpart of classical theorems on cycles of consecutive lengths due to Bondy and Bollobás in spectral graph theory, Nikiforov proposed the following open problem in 2008: What is the maximum C $C$ such that for all positive ε < C $\varepsilon \lt C$ and sufficiently large n $n$, every graph G $G$ of order n $n$ with spectral radius ρ( G ) > MathClass-open⌊ n 2 4 MathClass-close⌋ $\rho (G)\gt \sqrt{\lfloor \frac{{n}^{2}}{4}\rfloor }$ contains a cycle of length ℓ $\ell $ for each integer ℓ ∈[3 ,(C − ε ) n ] $\ell \in [3,(C-\varepsilon )n]$. We prove that C ≥ 1 4 $C\ge \frac{1}{4}$, improving the existing bounds. Besides several novel ideas, our proof technique is partly inspired by the recent research on Ramsey numbers of star versus large even cycles due to Allen, Łuczak, Polcyn, and Zhang, and with aid of a powerful spectral inequality. We also derive an Erdős–Gallai‐type edge number condition for even cycles, which may be of independent interest.

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Section: Note Addedmentioning

confidence: 99%

Eigenvalues and cycles of consecutive lengths

Ning

2023

Journal of Graph Theory

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The maximum number of cliques in graphs with bounded odd circumference

Cited by 1 publication

References 0 publications

Eigenvalues and cycles of consecutive lengths

Eigenvalues and cycles of consecutive lengths

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