On the strength of connectedness of unions of random graphs
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915790079393792 |
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| author | Bloznelis, Mindaugas |
| author_facet | Bloznelis, Mindaugas |
| contents | 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 $a=\min\{t\ge 1:\, {\bf P}\{δ(G_1)=t>0\}\}$ be the minimal non zero vertex degree attained with positive probability. Given $k\ge 0$ let $λ(k)=\ln n+k\ln\frac{m}{n}-\frac{m}{n} {\bf E} X$, 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 $λ(k)\to -\infty$, and ${\bf P}\{\cup_{i=1}^mG_i$ is $ak+1$-connected$\} \to 0 $ for $λ(k)\to +\infty$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02166 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the strength of connectedness of unions of random graphs Bloznelis, Mindaugas Combinatorics Probability 05C80, 05C82 G.2.2 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 $a=\min\{t\ge 1:\, {\bf P}\{δ(G_1)=t>0\}\}$ be the minimal non zero vertex degree attained with positive probability. Given $k\ge 0$ let $λ(k)=\ln n+k\ln\frac{m}{n}-\frac{m}{n} {\bf E} X$, 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 $λ(k)\to -\infty$, and ${\bf P}\{\cup_{i=1}^mG_i$ is $ak+1$-connected$\} \to 0 $ for $λ(k)\to +\infty$. 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. |
| title | On the strength of connectedness of unions of random graphs |
| topic | Combinatorics Probability 05C80, 05C82 G.2.2 |
| url | https://arxiv.org/abs/2602.02166 |