On the strength of connectedness of unions of random graphs

Abstract

Let $G_1,\dots, G_m$ be independent identically distributed random subgraphs of the complete graph ${\cal K}_n$. We analyse the threshold behaviour of the strength of connectedness of the union $\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$. Let $a=\min\{t\ge 1:\, {\bf P}\{\delta(G_1)=t>0\}\}$ be the minimal non zero vertex degree attained with positive probability. Given $k\ge 0$ let $\lambda(k)=\ln n+k\ln\frac{m}{n}-\frac{m}{n} {\bf E} X$, where $X$ stands for the number of non isolated vertices of $G_1$. Letting $n,m\to+\infty$ we show that ${\bf P}\{\cup_{i=1}^mG_i$ is $a(k+1)$-connected$\} \to 1 $ for $\lambda(k)\to -\infty$, and ${\bf P}\{\cup_{i=1}^mG_i$ is $ak+1$-connected$\} \to 0 $ for $\lambda(k)\to +\infty$. In particular, the connectivity strength of the union graph $\cup_{i=1}^mG_i$ increases in steps of size $a$. Our results are obtained in a more general setting where the contributing random subgraphs do not need to be identically distributed.