2,349 research outputs found
Graph coloring with no large monochromatic components
For a graph G and an integer t we let mcc_t(G) be the smallest m such that
there exists a coloring of the vertices of G by t colors with no monochromatic
connected subgraph having more than m vertices. Let F be any nontrivial
minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any
n-vertex graph G \in F. This bound is asymptotically optimal and it is attained
for planar graphs. More generally, for every such F and every fixed t we show
that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G
with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}).
It is also interesting to consider graphs of bounded degrees. Haxell, Szabo,
and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5.
We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n),
and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex
graphs of maximum degree 7, average degree at most 6+\epsilon for all
subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only
that the maximum order of magnitude of \mcc_2 is between \sqrt n and n.
We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure
Tunneling behavior of Ising and Potts models in the low-temperature regime
We consider the ferromagnetic -state Potts model with zero external field
in a finite volume and assume that the stochastic evolution of this system is
described by a Glauber-type dynamics parametrized by the inverse temperature
. Our analysis concerns the low-temperature regime ,
in which this multi-spin system has stable equilibria, corresponding to the
configurations where all spins are equal. Focusing on grid graphs with various
boundary conditions, we study the tunneling phenomena of the -state Potts
model. More specifically, we describe the asymptotic behavior of the first
hitting times between stable equilibria as in probability,
in expectation, and in distribution and obtain tight bounds on the mixing time
as side-result. In the special case , our results characterize the
tunneling behavior of the Ising model on grid graphs.Comment: 13 figure
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