Turán numbers of cycles plus a general graph

Abstract

For a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-free if it does not contain a member of $\mathcal{F}$ as a subgraph. The Turán number $\text{ex}(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph which is $\mathcal{F}$-free. Let ${\mathcal{C}}_{\ge k}$ be the set of cycles with length at least $k$. In this paper, we investigate the Turán number of $\{{\mathcal{C}}_{\ge k},F\}$ for a general graph $F$. To be precise, we determine $\text{ex}(n,\{{\mathcal{C}}_{\ge k},F\})$ 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 Turán number of $\{{\mathcal{C}}_{\ge k},K_r\}$ by the first author, Ning, and the third author.

corrected typos