Random Perfect Graphs

Colin McDiarmid; Nikola Yolov

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arXiv · 1604.00890

Random Perfect Graphs

Colin McDiarmid·Nikola Yolov

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Abstract

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number $α(P_n)$ and clique number $ω(P_n)$ is close to a concentrated distribution $L(n)$ which plays an important role in our generation method. We also prove that the probability that $P_n$ contains any given graph $H$ as an induced subgraph is asymptotically $0$ or $\frac12$ or $1$. Further we show that almost all perfect graphs are $2$-clique-colourable, improving a result of Bacsó et al from 2004; they are almost all Hamiltonian; they almost all have connectivity $κ(P_n)$ equal to their minimum degree; they are almost all in class one (edge-colourable using $Δ$ colours, where $Δ$ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon $W_P(x, y) = \frac12(\mathbb{1}[x \le 1/2] + \mathbb{1}[y \le 1/2])$.

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BibTeX RIS

Colin McDiarmid, Nikola Yolov. 2017-09-05. Random Perfect Graphs. https://arxiv.org/abs/1604.00890

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