Connected Tur\'an number of trees

TL;DR

This paper introduces the connected Turán number for trees, compares it to the classical Turán number, and determines exact values for small trees, advancing understanding of extremal connected graphs avoiding trees.

Contribution

It defines the connected Turán number for trees, analyzes its relation to the classical Turán number, and provides exact values for small trees, offering new constructions based on graph parameters.

Findings

Exact values of $ex_c(n,T)$ for trees with up to six vertices.

The ratio of $ex_c(n,T)$ to $(|T|-2)n/2$ can be significantly smaller than 1.

New constructions of connected $T$-free graphs based on graph parameters.

Abstract

As a variant of the much studied Tur\'an number, $ex(n,F)$, the largest number of edges that an $n$-vertex $F$-free graph may contain, we introduce the connected Tur\'an number $ex_c(n,F)$, the largest number of edges that an $n$-vertex connected $F$-free graph may contain. We focus on the case where the forbidden graph is a tree. The celebrated conjecture of Erd\H{o}s and S\'os states that for any tree $T$, we have $ex(n,T)\le(|T|-2)\frac{n}{2}$. We address the problem how much smaller $ex_c(n,T)$ can be, what is the smallest possible ratio of $ex_c(n,T)$ and $(|T|-2)\frac{n}{2}$ as $|T|$ grows. We also determine the exact value of $ex_c(n,T)$ for small trees, in particular for all trees with at most six vertices. We introduce general constructions of connected $T$-free graphs based on graph parameters as longest path, matching number, branching number, etc.