Tur\'an numbers of cycles plus a general graph

TL;DR

This paper determines the maximum number of edges in large graphs that avoid certain cycle lengths and a specified graph, extending previous Turán number results to more general graph families.

Contribution

It extends known Turán number results to include families with cycles of length at least k and a general graph F, under specific connectivity and bipartiteness conditions.

Findings

Precisely determined Turán numbers for these graph families.

Extended previous results from complete graphs to general graphs.

Provided bounds up to a constant additive term.

Abstract

For a family of graphs $\cal F$, a graph $G$ is $\cal F$-free if it does not contain a member of $\cal F$ as a subgraph. The Tur\'an number $\textrm{ex}(n,{\cal F})$ is the maximum number of edges in an $n$-vertex graph which is $\cal F$-free. Let ${\cal C}_{\geq k}$ be the set of cycles with length at least $k$. In this paper, we investigate the Tur\'an number of $\{{\cal C}_{\geq k}, F\}$ for a general graph $F$. To be precise, we determine $\textrm{ex}(n, \{{\cal C}_{\geq k}, F\})$ apart from a constant additive term, where $F$ either is a 2-connected nonbipartite graph or is a 2-connected bipartite graph under some conditions. This is an extension of a previous result on the Tur\'an number of $\{{\cal C}_{\geq k}, K_r\}$ by the first author, Ning, and the third author.