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For a graph , let denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that for every graph G, , where is the maximum size of an independent set of G. Erdős conjectured in the 80s that for almost every graph G equality holds, that is that for the random graph ...
Let Δ > 1 be a fixed positive integer. For
\documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\textbf{ {z}}} \in \mathbb{R}_+^\Delta\end{align*} \end{document}
let Gz be chosen uniformly at random from the collection of graphs on ∥z∥1n vertices that have zin vertices of degree i for i = 1,…,Δ. We determine the likely evolution...
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