Coloring random graphs online without creating monochromatic subgraphs

TL;DR

This paper studies an online coloring process for random graphs, establishing a threshold based on a new deterministic game concept called online vertex-Ramsey density, which is computable and guides when monochromatic subgraphs are avoided.

Contribution

It introduces the online vertex-Ramsey density as a key parameter for threshold determination and proves its computability, providing algorithms for successful coloring below the threshold.

Findings

Threshold function for online coloring is n^{-1/m_1^*(F,r)}.

Online vertex-Ramsey density is a rational number.

Polynomial-time algorithms succeed below the threshold.

Abstract

Consider the following random process: The vertices of a binomial random graph $G_{n,p}$ are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number $r$ of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph $F$ in the process. Our first main result is that for any $F$ and $r$, the threshold function for this problem is given by $p_0(F,r,n)=n^{-1/m_1^*(F,r)}$, where $m_1^*(F,r)$ denotes the so-called \emph{online vertex-Ramsey density} of $F$ and $r$. This parameter is defined via a purely deterministic two-player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any $F$ and $r$, the online vertex-Ramsey density $m_1^*(F,r)$ is a computable rational number. Our lower bound proof is algorithmic, i.e., we obtain polynomial-time online algorithms that succeed in coloring $G_{n,p}$ as desired with probability $1-o(1)$ for any $p(n) = o(n^{-1/m_1^*(F,r)})$.