Rooted induced trees in triangle-free graphs

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 ∈ V (G), let t(G, v) 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(G, v), respectively) over all connected triangle-free graphs G (and vertices v ∈ V (G)) on n vertices is denoted by t3(n) (t∗3(n)). Clearly, t(G, v) ≤ t(G) for all v ∈ V (G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that G is connected and triangle-free. We show that |G| ≤ 1 + (t(G,v)−1)t(G,v) 2 and determine the unique extremal graphs. Thus, we get as corollary that t3(n) ≥ t∗3(n) = ⌈ 12 (1 + √ 8n − 7)⌉, improving a recent result by Fox, Loh and Sudakov. All graphs in this note are simple and finite. For notation not defined here we refer the reader to Diestel’s book [1]. For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. The problem of bounding t(G) was first studied by Erdős, Saks and Sós [2] for certain classes of graphs, one of them being triangle-free graphs. Let t3(n) be the minimum of t(G) over all connected triangle-free graphs G on n vertices. Erdős, Saks and Sós showed that