Component Games on Random Graphs

TL;DR

This paper investigates the component game on random graphs, revealing a strong link between game outcomes and the emergence of a specific core structure, and establishes thresholds for Maker and Breaker strategies based on the graph's core properties.

Contribution

It connects the component game outcome on random graphs to the appearance of the (b+2)-core and provides thresholds for Maker and Breaker strategies based on this core's emergence.

Findings

Maker can build a linear-size component if c > c_{b+2}

Breaker can prevent large components if c < c_{b+2}

After constant iterations, the graph fragments into small pieces

Abstract

In the $\left(1:b\right)$ component game played on a graph $G$, two players, Maker and Breaker, alternately claim~$1$ and~$b$ previously unclaimed edges of $G$, respectively. Maker's aim is to maximise the size of a largest connected component in her graph, while Breaker is trying to minimise it. We show that the outcome of the game on the binomial random graph is strongly correlated with the appearance of a nonempty $(b+2)$-core in the graph. For any integer $k$, the $k$-core of a graph is its largest subgraph of minimum degree at least $k$. Pittel, Spencer and Wormald showed in 1996 that for any $k\ge3$ there exists an explicitly defined constant $c_{k}$ such that $p=c_{k}/n$ is the threshold function for the appearance of the $k$-core in $G(n,p)$. More precisely, $G(n,c/n)$ has WHP a linear-size $k$-core when the constant $c>c_{k}$, and an empty $k$-core when $c<c_{k}$. We show that for any positive constant $b$, when playing the $(1:b)$ component game on $G(n,c/n)$, Maker can WHP build a linear-size component if $c>c_{b+2}$, while Breaker can WHP prevent Maker from building larger than polylogarithmic-size components if $c<c_{b+2}$. For Breaker's strategy, we prove a theorem which may be of independent interest. The standard algorithm for computing the $k$-core of any graph is to repeatedly delete ("peel") all vertices of degree less than $k$, as long as such vertices remain. When $G(n,c/n)$ for $c<c_{k}$, it was shown by Jiang, Mitzenmacher and Thaler that $\log_{k-1}\log n+\Theta(1)$ peeling iterations are WHP necessary and sufficient to obtain the (empty) $k$-core of~$G$. Our theorem states that already after a constant number of iterations, $G$ is WHP shattered into pieces of polylogarithmic size.