Rooted induced trees in triangle-free graphs

TL;DR

This paper determines the maximum size of connected triangle-free graphs with a given rooted induced tree size, providing exact bounds and identifying unique extremal graphs, thus improving previous bounds on these parameters.

Contribution

It solves the extremal problem of maximizing graph size for a fixed rooted induced tree size in connected triangle-free graphs, with exact bounds and characterization of extremal graphs.

Findings

Established an upper bound: |G| ≤ 1 + ((t_v(G)-1)t_v(G))/2

Determined the unique extremal graphs achieving this bound

Improved the lower bounds for t_3(n) and t_3^v(n)

Abstract

For a graph $G$, let $t(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree. Further, for a vertex $v\in V(G)$, let $t^v(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree, with the extra condition that the tree must contain $v$. The minimum of $t(G)$ ($t^v(G)$, respectively) over all connected triangle-free graphs $G$ (and vertices $v\in V(G)$) on $n$ vertices is denoted by $t_3(n)$ ($t_3^v(n)$). Clearly, $t^v(G)\le t(G)$ for all $v\in V(G)$. In this note, we solve the extremal problem of maximizing $|G|$ for given $t^v(G)$, given that $G$ is connected and triangle-free. We show that $|G|\le 1+\frac{(t_v(G)-1)t_v(G)}{2}$ and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^v(n)=\lceil {1/2}(1+\sqrt{8n-7})\rceil$, improving a recent result by Fox, Loh and Sudakov.