On large induced trees and long induced paths in sparse random graphs

W.C.Stephen Suen
1992 Journal of combinatorial theory. Series B (Print)  
Let G" denote the graph obtained from deleting the edges of K", the complete graph with vertex set V" = (1,2, . . . . n}, independently with equal probability 1 -p. Assume that p = p(n) is such that np = c > 1. We describe an algorithm FindTree for finding induced trees in a graph. By analyzing how FindTree performs in Gn,p, we obtain the following results. Let T" be the order of the largest induced subtree of G, p' We find a number t(c) such that T" is almost surely larger than (t(c) -&)n for
more » ... ny E > 0. Also, if L, denotes the length of the longest induced path in Gn,p, then we find a number h(c) such that L, is almost surely larger than (h(c) -E)n for any E>O.
doi:10.1016/0095-8956(92)90021-o fatcat:phkhtc3m65gexiuvt6kc3u3so4