On the phase transitions of graph coloring and independent sets

Details On the phase transitions of graph coloring and independent sets On the phase transitions of graph coloring and independent sets

2003-09-23

arxiv.org/abs/cond-mat/0309518v1

Physica A 343 (2004), 401-423

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

10.1016/j.physa.2004.05.055

We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are `shortest' peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the `widest' acyclic orientations and the independence number.

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