Rooted induced trees in triangle-free graphs

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)...

Induced Trees in Triangle-Free Graphs

We prove that every connected triangle-free graph on n vertices contains an induced tree on exp(c √ log n ) vertices, where c is a positive constant. The best known upper bound is (2 + o(1)) √ n. This partially answers questions of Erdős, Saks, and Sós and of Pultr.

Maximal Induced Matchings in Triangle-Free Graphs

An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. It is known that every n-vertex graph has at most 10 ≈ 1.5849 maximal induced matchings, and this bound is best possible. We prove that every n-vertex triangle-free graph has at most 3 ≈ 1.4423 maximal induced matchings, and this bound is attained by every disjoint union of copies of the complete bipar...

Large induced trees in Kr-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. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph Kr on r vertices. This problem was posed twenty years ago by Erdős, Saks, and Sós. Substantially improving earlier results of various researchers, we prove that every connected triangle-fre...

Measuring triangle-free graphs