On Coloring Random Subgraphs of a Fixed Graph

Given an arbitrary graph \(G\) we study the chromatic number of a random subgraph \(G_{1/2}\) obtained from \(G\) by removing each edge independently with probability \(1/2\). Studying \(\chi(G_{1/2})\) has been suggested by Bukh~\cite{Bukh}, who asked whether \(\mathbb{E}[\chi(G_{1/2})] \geq \Omega( \chi(G)/\log(\chi(G)))\) holds for all graphs \(G\). In this paper we show that for any graph \(G\) with chromatic number \(k = \chi(G)\) and for all \(d \leq k^{1/3}\) it holds that \(\Pr[\chi(G_{1/2}) \leq d] < \exp \left(- \Omega\left(\frac{k(k-d^3)}{d^3}\right)\right)\). In particular, \(\Pr[G_{1/2} \text{ is bipartite}] < \exp \left(- \Omega \left(k^2 \right)\right)\). The later bound is tight up to a constant in \(\Omega(\cdot)\), and is attained when \(G\) is the complete graph on \(k\) vertices. As a technical lemma, that may be of independent interest, we prove that if in \emph{any} \(d^3\) coloring of the vertices of \(G\) there are at least \(t\) monochromatic edges, then \(\Pr[\chi(G_{1/2}) \leq d] < e^{- \Omega\left(t\right)}\). We also prove that for any graph \(G\) with chromatic number \(k = \chi(G)\) and independence number \(\alpha(G) \leq O(n/k)\) it holds that \(\mathbb{E}[\chi(G_{1/2})] \geq \Omega \left( k/\log(k) \right)\). This gives a positive answer to the question of Bukh for a large family of graphs.