Extremal problems for a matching and any other graph

TL;DR

This paper determines the maximum number of cliques in large graphs that avoid a specific matching and other subgraphs, extending classical Turán-type extremal results with exact formulas under certain conditions.

Contribution

It provides exact and asymptotic values for the generalized Turán number for graphs avoiding a matching and an arbitrary graph, including non-bipartite and color-critical cases.

Findings

Exact value of extremal number for non-bipartite graphs.

Asymptotic bounds for bipartite graphs.

Results extend classical Turán theorems.

Abstract

For a family of graphs $\F$, a graph is called $\F$-free if it does not contain any member of $\F$ as a subgraph. The generalized Tur\'an number $\ex(n,K_r,\F)$ is the maximum number of $K_r$ in an $n$-vertex $\F$-free graph and $\ex(n,K_2,\F)=\ex(n,\F)$, i.e., the classical Tur\'an number. Let $M_{s+1}$ be a matching on $s+1$ edges and $F$ be any graph. In this paper, we determine $\ex(n,K_r, \{M_{s+1},F\})$ apart from a constant additive term and also give a condition when the error constant term can be determined. In particular, we give the exact value of $\ex(n,\{M_{s+1},F\})$ for $F$ being any non-bipartite graph or some bipartite graphs. Furthermore, we determine $\ex(n,K_r,\{M_{s+1},F\})$ when $F$ is color critical with $\chi(F)\ge \max\{r+1,4\}$. These extend the results in [2,11,18].