1 research outputs found
Expected Chromatic Number of Random Subgraphs
Given a graph and , let denote the random subgraph of
obtained by keeping each edge independently with probability . Alon,
Krivelevich, and Sudokov proved , and Bukh conjectured an improvement of
. We prove a new
spectral lower bound on , as progress towards Bukh's
conjecture. We also propose the stronger conjecture that for any fixed , among all graphs of fixed chromatic number, is
minimized by the complete graph. We prove this stronger conjecture when is
planar or . We also consider weaker lower bounds on
proposed in a recent paper by Shinkar; we answer two
open questions of Shinkar negatively and propose a possible refinement of one
of them.Comment: 8 pages plus an appendix (14 total). Multiple figures, one of which
before the appendix. Work done as part of the 2018 program of the Summer
Undergraduate Math Research at Yal