Rainbow Turán problems for a matching and any other graph
Dániel Gerbner, Shujing Miao
TL;DR
This work develops a rainbow Turán framework for collections of graphs on a common vertex set, focusing on maximizing edge resources while avoiding rainbow copies of a forbidden family ${F, M_{s+1}}$. It introduces three extremal parameters $\mathrm{ex}_t(n,{\mathcal{F}})$, $\mathrm{ex}_t^{\sum}(n,{\mathcal{F}})$, and $\mathrm{ex}_t^{\prod}(n,{\mathcal{F}})$ and derives exact or asymptotic formulas for ${\mathcal{F}}=\{F, M_{s+1}\}$ across four regimes determined by the bipartiteness of $F$ and the color-covering parameter $p(F)$. The proof strategy combines greedy rainbow-extension arguments, a nesting reduction to structured colorings, and Hall-type arguments via an auxiliary color-vertex graph to bound the sum and product quantities, yielding explicit constructions and tight bounds. The results extend rainbow Turán-type theory beyond triangles to general $F$, clarifying how matching constraints interact with forbidden subgraphs and revealing when rainbow-extremal configurations are compositions of complete-graph colors and balanced bipartite blocks. Overall, the paper provides a cohesive, extensible toolkit for multicolor extremal problems with rainbow restrictions and connects to Erdős–Sós-type questions in the bipartite and tree regimes.
Abstract
For a family of graphs $\cF$, a graph is called $\cF$-free if it does not contain any member of $\cF$ as a subgraph. Given a collection of graphs $(G_1,\ldots,G_t)$ on the same vertex set $V$ of size $n$, a rainbow graph on $V$ is obtained by taking at most one edge from each $G_i$. We say that a collection is rainbow $\cF$-free if it contains no rainbow copy of any member of $\cF$. In this paper, we study the maximum values of $min_{i\in [t]}|E(G_i)|$, $\sum_{i=1}^{t}|E(G_i)|$ and $\prod_{i=1}^{t}|E(G_i)|$ among rainbow $\{F,M_{s+1}\}$-free collections $(G_1,\ldots,G_t)$ on $n$ vertices.