Large induced trees in Kr -free graphs

Authors:
Jacob Fox;Po-Shen Loh;Benny Sudakov
Affiliations:
Department of Mathematics, Princeton University, Princeton, NJ 08544, United States;Department of Mathematics, Princeton University, Princeton, NJ 08544, United States;Department of Mathematics, UCLA, Los Angeles 90095, United States
Venue:
Journal of Combinatorial Theory Series B
Year:
2009

Citing 8
Cited 1

Maximum induced trees in graphs

Journal of Combinatorial Theory Series B
On the order of the largest induced tree in a random graph

Discrete Applied Mathematics
Large induced trees in sparse random graphs

Journal of Combinatorial Theory Series B
Maximal induced trees in sparse random graphs

Discrete Mathematics - First Japan Conference on Graph Theory and Applications
On large induced trees and long induced paths in sparse random graphs

Journal of Combinatorial Theory Series B
Radius two trees specify &khgr;-bounded classes

Journal of Graph Theory
The largest induced tree in a sparse random graph

Proceedings of the seventh international conference on Random structures and algorithms
Induced trees in graphs of large chromatic number

Journal of Graph Theory

Subgraphs of Weakly Quasi-Random Oriented Graphs

SIAM Journal on Discrete Mathematics

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Abstract

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 K"r on r vertices. This problem was posed twenty years ago by Erdos, Saks, and Sos. Substantially improving earlier results of various researchers, we prove that every connected triangle-free graph on n vertices contains an induced tree of order n. When r=4, we also show that t(G)=logn4logr for every connected K"r-free graph G of order n. Both of these bounds are tight up to small multiplicative constants, and the first one disproves a recent conjecture of Matousek and Samal.