On the Tur\'an number of double stars

TL;DR

This paper investigates the maximum edges in large graphs avoiding a specific double star structure, extending known results for smaller cases to more complex configurations.

Contribution

It determines the extremal graphs for the Turán number of double stars with parameters (3, b), revealing more intricate structures than previous cases.

Findings

Explicit characterization of extremal graphs for $ex(n,S_{3,b})$

Extension of Turán number results to more complex double star graphs

Identification of structural differences from simpler double star cases

Abstract

The Tur\'an number of a graph $F$, $ex(n,F)$, is the maximum number of edges in a graph on $n$ vertices which does not contain $F$ as a subgraph. Let $S_{a,b}$ denote a double star with a central edge $uv$, $a$ leaves connected to $u$ and $b$ leaves connected to $v$. The function $ex(n,S_{a,b})$ has been studied for $a=1,2$, their extremal graphs are disjoint copies of $K_{a+b+1}$ and either a small clique or a near $b$-regular graph. In this paper, we further study $ex(n,S_{3,b})$ and determine the extremal graphs, which have more structures than those of $a=1,2$.