Tur\'an's Problem for Trees

TL;DR

This paper determines the maximum number of edges in graphs of a given size that do not contain specific trees, providing exact extremal values for two particular tree structures.

Contribution

It offers exact extremal edge counts for graphs avoiding two specific trees, expanding understanding of Turán's problem for trees.

Findings

Exact values of ex(p;T_n) are established.

Exact values of ex(p;T_n^*) are established.

Results contribute to extremal graph theory for tree-avoiding graphs.

Abstract

For a forbidden graph $L$, let $ex(p;L)$ denote the maximal number of edges in a simple graph of order $p$ not containing $L$. Let $T_n$ denote the unique tree on $n$ vertices with maximal degree $n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$ and $E=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-3}v_{n-2},v_{n-2}v_{n-1}\}$. In the paper we give exact values of $ex(p;T_n)$ and $ex(p;T_n^*)$.