On MAXCUT in strictly supercritical random graphs, and coloring of random graphs and random tournaments

TL;DR

This paper analyzes the structure and properties of strictly supercritical random graphs, determining typical MAXCUT sizes, disproving a homomorphism conjecture, and exploring coloring behaviors of biased random tournaments.

Contribution

It applies a structural theorem to find MAXCUT sizes in supercritical random graphs and proves a conjecture on graph homomorphisms, also analyzing coloring in biased random tournaments.

Findings

MAXCUT size in G(n,(1+ε)/n) is characterized in terms of ε

G(n,(1+ε)/n) is not homomorphic to certain odd cycles with high probability

Chromatic number of p-random tournaments with p=Θ(1/n) behaves like that of a random graph

Abstract

We use a theorem by Ding, Lubetzky and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of $G\sim G\left(n,\frac {1+\varepsilon}n\right)$ in terms of $\varepsilon$. We then apply this result to prove the following conjecture by Frieze and Pegden. For every $\varepsilon>0$ there exists $\ell_\varepsilon$ such that \whp $G\sim G(n,\frac {1+\varepsilon}n)$ is not homomorphic to the cycle on $2\ell_\varepsilon+1$ vertices. We also consider the coloring properties of biased random tournaments. A $p$-random tournament on $n$ vertices is obtained from the transitive tournament by reversing each edge independently with probability $p$. We show that for $p=\Theta(\frac 1n)$ the chromatic number of a $p$-random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case $p=\frac {1+\varepsilon}n$ we use the aforementioned result on MAXCUT.