On Turán problems with bounded matching number
Dániel Gerbner
TL;DR
This work addresses Turán-type extremal problems for $n$-vertex graphs that avoid a fixed graph $F$ while having bounded matching number $s$. It generalizes prior results by proving that, for $\chi(F)>2$ and large $n$, the extremal number satisfies $$\mathrm{ex}(n,\{F,M_{s+1}\})=\mathrm{ex}(s,\mathcal{F})+s(n-s)$$ with $\mathcal{F}$ the family of graphs obtained from $F$ by deleting an independent set, and provides exact or near-exact results in the bipartite setting when $p(F)\le s$. The proofs combine a constructive lower bound via attaching $n-s$ universal vertices to an $\mathcal{F}$-free $s$-vertex graph with a sharp degree-based upper bound, yielding a tight edge-count of $\text{ex}(s,\mathcal{F})+s(n-s)$, along with detailed analysis of extremal structures for trees, paths, and two proposed constructions. Overall, the results advance our understanding of how forbidding both $F$ and a large matching constrains edge density, and provide concrete extremal graphs and asymptotics for a broad class of forbidden-configurations problems.
Abstract
Very recently, Alon and Frankl initiated the study of the maximum number of edges in $n$-vertex $F$-free graphs with matching number at most $s$. For fixed $F$ and $s$, we determine this number apart from a constant additive term. We also obtain several exact results.