We study a simple Markov chain, known as the Glauber dynamics, for generating a random <i>k</i>-coloring of a <i>n</i>-vertex graph with maximum degree Δ. We prove that the dynamics converges to a random coloring after <i>O</i>(<i>n</i> log <i>n</i>) steps assuming <i>k</i> ≥ <i>k</i><sub>0</sub> for some absolute constant <i>k</i><sub>0</sub>, and either: ... more >>>
The Tutte-Gr\"othendieck polynomial $T(G;x,y)$ of a graph $G$
encodes numerous interesting combinatorial quantities associated
with the graph. Its evaluation in various points in the $(x,y)$
plane give the number of spanning forests of the graph, the number
of its strongly connected orientations, the number of its proper
$k$-colorings, the (all ...
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