Connectivity of Random Cubic Sum Graphs

Abstract

Consider the set $\mathcal{SG}(Q_k)$ of all graphs whose vertices are labeled with nonidentity elements of the group $Q_k=\mathbb{Z}_2^k$ so that there is an edge between vertices with labels a and b if and only if the vertex labeled $a+b$ is also in the graph. Note that edges always appear in triangles since $a+b=c$, $b+c=a$, and $a+c=b$ are equivalent statements for $Q_k$. We define the random cubic sum graph $\mathcal{SG}(Q_k,p)$ to be the probability space over $\mathcal{SG}(Q_k)$ whose vertex sets are determined by $\Pr[x\in V]=p$ with these events mutually independent. As p increases from 0 to 1, the expected structure of $\mathcal{SG}(Q_k,p)$ undergoes radical changes. We obtain thresholds for some graph properties of $\mathcal{SG}(Q_k,p)$ as $k\rightarrow\infty$. As with the classical random graph, the threshold for connectivity coincides with the disappearance of the last isolated vertex.