Turán problems for suspension of a balanced tree

TL;DR

This work determines sharp Turán-type bounds for the suspension $\widehat{T}$ of a balanced tree $T$ in terms of a function $f(n,k)$, under the Erdős-Sós conjecture, for large $n$ (specifically $n\ge 4(4k)^6$). The authors develop a decomposition-family framework and a minimum-degree reduction to prove $\ex(n,\widehat{T})\le f(n,k)$ and characterize extremal graphs, with tightness shown for infinitely many $n$ and full sharpness for trees $T$ containing a matching of size $k$. The results generalize known cases, introduce a new class of graphs whose decomposition family does not include a linear forest, and provide constructions achieving equality, along with conditions under which equality holds for all $n$. This advances understanding of Turán problems for sparse graphs via suspension and decompositional methods, broadening the class of tractable extremal problems beyond linear-forest decompositions.

Abstract

The Turán number $\ex(n,H)$ is the maximum number of edges that an $n$-vertex $H$-free graph can have. The suspension $\widehat{H}$ is obtained from $H$ by adding a new vertex which is adjacent to all vertices of $H$ and a tree is balanced if the sizes of its two color classes differ at most $1$. In this paper, we obtain a sharp bound of $\ex(n,\widehat{T})$ when $n\ge 4(4k)^6$ based on the Erdős-Sós conjecture. We also show the bound is sharp for infinitely many $n$ and characterize all extremal graphs. In particular, if $T$ satisfies some conditions such as $T$ contains a matching covering all vertices in one color class, then the bound is sharp for all $n$. This is a new class of graphs whose decomposition family does not contain a linear forest but we still can determine its Turán number.

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Theorems & Definitions (20)

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