The number of 4-cycles in a graph

Ting Li and Han Ren

Applied Mathematics and Computation, 2022, vol. 428, issue C

Abstract: In extremal graph theory studies, it is not yet finished to determine the exact value of ex(n;C4) in general case. Erdős and Simonovits [10] conjectured that an n-vertex graph G with ex(n;C4)+1 edges may contain at least n+o(n) 4-cycles. In this paper we investigate the number of 4-cycles in a graph. We show that the number of 4-cycles in an n-vertex graph G is at least (ϵ4n−3)/4+ϵ2/(2n) provided that |E(G)|=n(1+4n−3)/4+ϵ, where ϵ is an arbitrary positive real number. This result is an approach solving to this open problem. Furthermore, we prove that if n is large enough, then there are at least ((4ϵ−1)4n−3+5)/16+o(1) 4-cycles in an n-vertex graph with e=(n−1)(1+4n−3)/4+ϵ edges which has k vertices whose degrees are mutually different, where k≥24n3+12n−(12n)2−1/273 and ϵ>0 is a real number. In addition, we get that a triangle-free graph with n vertices and e=nn−1/2+ϵ edges has at least (2ϵ(n−1−2)+1)/4+o(1) 4-cycles, where ϵ>0 is an any real number and n is large enough. This shows that the weak version of Erdős and Simonovits' conjecture [10] holds for such graphs when n is sufficiently large. Finally, we obtain a bipartite version of this result.

Keywords: Turán number; Erdős’ conjecture; 4-cycles (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300322002855
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:428:y:2022:i:c:s0096300322002855

DOI: 10.1016/j.amc.2022.127211

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().