Generalized Turán problem with bounded matching number
Yue Ma, Xinmin Hou
TL;DR
The paper studies the generalized Turán number $ex(n, T, {K_{k+1}, M_{s+1}})$, focusing on counting copies of $T$ in $n$-vertex graphs free of a $K_{k+1}$-clique and a matching of size $s+1$. Extending prior work that treated $T=K_2$, the authors determine the exact value of $ex(n, K_r, {K_{k+1}, M_{s+1}})$ for $r \ge 3$ and obtain an $O(1)$-error result for general $H$ with $ex(n, K_r, {H, M_{s+1}})$. The core approach combines the Tutte–Berge decomposition, Zykov-type symmetrization, and the switching method to reduce the extremal graphs to two canonical constructions: $T_k(2s+1)$ and $G_k(n,s)=\overline{K}_{n-s} \vee T_{k-1}(s)$, yielding $ex(n, K_r, {K_{k+1}, M_{s+1}}) = \max\{ N(T_k(2s+1), K_r), N(G_k(n,s), K_r) \}$. A parallel result for stars $S_r$ is obtained under the condition $n \ge 2(s+1)(r+1)$, and the authors discuss limitations and conjectures for broader ranges of $n$ and $r$ based on observed counterexamples. The work thus provides explicit extremal structures and sharp thresholds for generalized Turán problems with bounded matching, using a robust toolkit applicable to related forbidden-subgraph settings.
Abstract
For a graph $T$ and a set of graphs $\mathcal{H}$, let $\mbox{ex}(n,T,\mathcal{H})$ denote the maximum number of copies of $T$ in an $n$-vertex $\mathcal{H}$-free graph. Recently, Alon and Frankl~(arXiv2210.15076) determined the exact value of $\mbox{ex}(n,K_2,\{K_{k+1},M_{s+1}\})$, where $K_{k+1}$ and $M_{s+1}$ are complete graph on $k+1$ vertices and matching of size $s+1$, respectively. Soon after, Gerbner~(arXiv2211.03272) continued the study by extending $K_{k+1}$ to general fixed graph $H$. In this paper, we continue the study of the function $\mbox{ex}(n, T,\{H,M_{s+1}\})$ when $T=K_r$ for $r\ge 3$. We determine the exact value of $\mbox{ex}(n,K_r,\{K_{k+1},M_{s+1}\})$ and give the value of $\mbox{ex}(n,K_r,\{H,M_{s+1}\})$ for general $H$ with an error term $O(1)$.