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 ve 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 is an element of V(G)) on n vertices is denoted by t(3)(n) (t(3)*(n)). Clearly, t(G, v) <= t(G) for all ye 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| <= + (t(G,v)-1t(G,v)/2 and determine the unique extremal graphs. Thus, we get as corollary that t(3)(n)>= t(3)*(n), [1/2(1+ root 8n-7)], improving a recent result by Fox, Loh and Sudakov. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 64: 206-209, 2010