On Coloring Random Subgraphs of a Fixed Graph

TL;DR

This paper investigates the chromatic number of random subgraphs formed by edge removal, providing probabilistic bounds and confirming a conjecture for certain graph families.

Contribution

It establishes bounds on the probability that the chromatic number drops below a threshold and confirms Bukh's conjecture for graphs with bounded independence number.

Findings

Probability that the chromatic number is small decreases exponentially with the original chromatic number.

Complete graphs are tight examples for the bounds on bipartiteness probability.

For graphs with small independence number, the expected chromatic number of the random subgraph is at least proportional to k/log(k).

Abstract

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.