Long paths and cycles in random subgraphs of graphs with large minimum degree

Combinatorics Seminar
Friday, February 22, 2013 - 3:00pm
1 hour (actually 50 minutes)
Skiles 005
For a given finite graph G of minimum degree at least k, let G_{p} be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if p \ge \omega/k for a function \omega=\omega(k) that tends to infinity as k does, then G_p asymptotically almost surely contains a cycle (and thus a path) of length at least (1-o(1))k, and (ii) if p \ge (1+o(1))\ln k/k, then G_p asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k+1 vertices. Joint w/ Michael Krivelevich (Tel Aviv), Benny Sudakov (UCLA).