Generalized Tur\'an results for disjoint copies of degenerate graphs

TL;DR

This paper extends previous results on the maximum number of copies of a graph H in an n-vertex graph avoiding multiple disjoint copies of another graph F, providing more precise bounds for new classes of forbidden graphs.

Contribution

It generalizes existing Turán-type results to broader classes of forbidden graphs, including disjoint unions of complete bipartite graphs and cycles, with sharper bounds.

Findings

Established bounds for x(n, K_r, (t+1)K_{a,b})

Derived bounds for x(n, K_r, (t+1)C_{2k})

Extended previous asymptotic results to new graph classes

Abstract

The generalized Tur\'{a}n number $\mathrm{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. For an integer $t \geq 1$, let $tF$ be the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku, and Vizer (2019) established an asymptotically sharp bound for $\mathrm{ex}(n,K_r,(t+1)K_{2,b})$. We extend their results in two directions by considering forbidden graphs $(t+1)K_{a,b}$ and $(t+1)C_{2k}$ and establish more precise matching upper and lower bounds of the same order of magnitude.