Tur\'an's problem for trees $T_n$ with maximal degree $n-4$

TL;DR

This paper determines the maximum number of edges in large graphs that do not contain specific trees with maximal degree n-4, providing explicit formulas for these extremal values.

Contribution

It derives explicit formulas for the extremal number of edges in graphs avoiding certain trees with maximal degree n-4, for all sufficiently large graphs.

Findings

Explicit formulas for ex(p;T_n^3), ex(p;T_n^{''}), ex(p;T_n^{'''}) are obtained.

Results apply to graphs with order p ≥ n ≥ 15.

The formulas characterize the extremal graphs avoiding these trees.

Abstract

For $n\ge 6$ let $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2}$, $v_1v_{n-1}\}$, $E_2=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2},v_2v_{n-1}\}$, $E_3=\{v_0v_1,\ldots,v_0v_{n-4}$, $v_1v_{n-3},v_2v_{n-2},v_3v_{n-1}\}$, $T_n^3=(V,E_1),\ T_n^{''}=(V,E_2)$ and $T_n^{'''} =(V,E_3).$ In this paper, for $p\ge n\ge 15$ we obtain explicit formulas for $ex(p;T_n^3)$, $ex(p;T_n^{''})$ and $ex(p;T_n^{'''})$, where $ex(p;L)$ denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph.