Clique counts under degree and matching constraints

Abstract

We study the maximum number of $t$-cliques in graphs with maximum degree at most $D$ and matching number at most $M$, with no restriction on the number of vertices. For $t\ge 3$, we determine this maximum whenever $D\ge 2M-1$ and characterize all extremal graphs when the maximum is positive. Up to isolated vertices, these are complete or complete split graphs for $D\ge 2M$, while $K_{2M}$ is uniquely extremal when $D=2M-1$. We also derive a general upper bound from the Chvátal-Hanson edge theorem and refine it for even $D$. For fixed even $D$ and $3\le t\le D+1$, every extremal graph consists, apart from isolated vertices, of copies of $K_{D+1}$ together with a remainder whose order is bounded independently of $M$. When $D/2$ divides $M$, this remainder is empty and the extremal graph is unique. This structure yields an eventual recurrence, reducing the determination of all extremal values and graphs to finitely many matching bounds.