Coloring Random Graphs Online Without Creating Monochromatic Subgraphs

Consider the following generalized notion of graph coloring: a coloring of the vertices of a graph~$G$ is \emph{valid} \wrt some given graph~$F$ if there is no copy of $F$ in $G$ whose vertices all receive the same color. We study the problem of computing valid colorings of the binomial random graph~$\Gnp$ on~$n$ vertices with edge probability~$p=p(n)$ in the following online setting: the vertices of an initially hidden instance of $\Gnp$ are revealed one by one (together with all edges leading to previously revealed vertices) and have to be colored immediately and irrevocably with one of $r$ available colors. It is known that for any fixed graph $F$ and any fixed integer $r\geq 2$ this problem has a threshold $p_0(F,r,n)$ in the following sense: For any function $p(n) = o(p_0)$ there is a strategy that \aas (asymptotically almost surely, i.e., with probability tending to 1 as $n$ tends to infinity) finds an $r$-coloring of $\Gnp$ that is valid \wrt $F$ online, and for any function~$p(n)=\omega(p_0)$ \emph{any} online strategy will \aas fail to do so. In this work we establish a general correspondence between this probabilistic problem and a 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. This characterization allows us to compute, for any $F$ and $r$, a value $\gamma=\gamma(F,r)$ such that the threshold of the probabilistic problem is given by $p_0(F,r,n)=n^{-\gamma}$. Our approach yields polynomial-time coloring algorithms that \aas find valid colorings of $\Gnp$ online in the entire regime below the respective thresholds, i.e., for any $p(n) = o(n^{-\gamma})$