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Extensions of the Erdős–Gallai theorem and Luo’s theorem

Part of: Graph theory

Published online by Cambridge University Press:  08 October 2019

Bo Ning  and
Xing Peng
Bo Ning
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin, 300062, P. R.China Emails: bo.ning@tju.edu.cn; x2peng@tju.edu.cn
Xing Peng*
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin, 300062, P. R.China Emails: bo.ning@tju.edu.cn; x2peng@tju.edu.cn
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Abstract

The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least

$${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$
edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

MSC classification

Primary: 05C35: Extremal problems 05C38: Paths and cycles
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 1 , January 2020 , pp. 128 - 136
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Supported by the NSFC grants (11601379, 11771141, 11971346) and the Seed Foundation of Tianjin University (2018XRG-0025).

‡

Supported by the NSFC grant (11601380) and the Seed Foundation of Tianjin University (2017XRX-0011)

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