Warmth and mobility of random graphs

Details Warmth and mobility of random graphs Warmth and mobility of random graphs

2010-09-04

arxiv.org/abs/1009.0792v2

Mathematics

Combinatorics

13 pages, 5 figures

Scientific paper

Brightwell and Winkler introduced the graph parameters warmth and mobility in the context of combinatorial statistical physics. They related both parameters to lower bounds on chromatic number. Although warmth is not a monotone graph property we show it is nevertheless "statistically monotone" in the sense that it tends to increase with added random edges, and that for sparse graphs ($p=O(n^{-\alpha})$, $\alpha > 0$) the warmth is concentrated on at most two values, and for most $p$ it is concentrated on one value. We also put bounds on warmth and mobility in the dense regime, and as a corollary obtain that a conjecture of Lov\'asz holds for almost all graphs.

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