Minimum Number of Monochromatic Subgraphs of a Random Graph
Yatin Dandi, David Gamarnik, Haodong Zhu
TL;DR
The paper addresses the problem of minimizing monochromatic copies of a fixed strictly 1-balanced graph $F$ in random graphs under a random 2-coloring. It leverages a contiguity between appearance of $F$ in $\mathbb{G}(n,p)$ and independent copies in the hypergraph model $\mathbb{H}_F(n,q)$ to establish the existence of a limit $m(F,c)$ for $p= c n^{-1/d_1(F)}$, and provides explicit asymptotics for $m(F,c)$ as the host graph size and $c$ grow. The analysis proceeds by translating the optimization into a mean-field spin-glass framework, where the monochromatic count is expressed via a Gaussianized Hamiltonian and a constrained variational problem. Through extremizing a family of Gaussians, the authors derive the leading asymptotics of the limit, yielding $m(F,c) = \kappa(F,c,s,r) + (1+o_r(1))\sqrt{(2\log 2)\kappa(F,c,s,r)} + o_c(c^{s/2})$ with $\kappa(F,c,s,r)= c^s/(2^{r-1}\mathrm{aut}(F))$. This provides a rigorous bridge between random graph theory and mean-field spin-glass methods, enabling precise predictions in sparse regimes.
Abstract
We consider the problem of minimizing the number of monochromatic subgraphs of a random graph, when each node of the host graph is assigned one of the two colors. Using a recently discovered contiguity between appearance of strictly balanced subgraphs $F$ in a random graph, and random hypergraphs where copies of $F$ are generated independently, we show that the minimum value converges to a limit, when the expected number of copies of $F$ is linear in the number of nodes $|V|$. Furthermore, using the connections with mean field spin glass models, we obtain an asymptotic expression for this limit as the normalized expected number of copies of $F$ and the size of $F$ diverge to infinity.