Generalized Tur\'an problems for double stars

TL;DR

This paper investigates the maximum number of edges in large graphs avoiding certain double star subgraphs, providing exact values and bounds for various generalized Turán problems involving double stars.

Contribution

It determines exact extremal functions for $ex(n,K_k,S_{a,b})$ and $ex(n,S_{a,b},F)$ for large $n$, and offers bounds on $ex(n,S_{a,b},S_{c,d})$, advancing Turán theory for double stars.

Findings

Exact values of $ex(n,K_k,S_{a,b})$ for large $n$

Exact values of $ex(n,S_{a,b},F)$ for specific graphs $F$

Bounds on $ex(n,S_{a,b},S_{c,d})$

Abstract

We study the generalized Tur\'an function $ex(n,H,F)$, when $H$ or $F$ is a double star $S_{a,b}$, which is a tree with a central edge $uv$, $a$ leaves connected to $u$ and $b$ leaves connected to $v$. We determine $ex(n,K_k,S_{a,b})$ and $ex(n,S_{a,b},F)$ for sufficiently large $n$, where $F$ is either a 3-chromatic graph with an edge whose deletion results in a bipartite graph, or the 2-fan, i.e. two triangles sharing a vertex. We also give bounds on $ex(n,S_{a,b},S_{c,d})$.