2,924 research outputs found
Lower bounds for on-line graph colorings
We propose two strategies for Presenter in on-line graph coloring games. The
first one constructs bipartite graphs and forces any on-line coloring algorithm
to use colors, where is the number of vertices in the
constructed graph. This is best possible up to an additive constant. The second
strategy constructs graphs that contain neither nor as a subgraph
and forces colors. The best known
on-line coloring algorithm for these graphs uses colors
Coloring half-planes and bottomless rectangles
We prove lower and upper bounds for the chromatic number of certain
hypergraphs defined by geometric regions. This problem has close relations to
conflict-free colorings. One of the most interesting type of regions to
consider for this problem is that of the axis-parallel rectangles. We
completely solve the problem for a special case of them, for bottomless
rectangles. We also give an almost complete answer for half-planes and pose
several open problems. Moreover we give efficient coloring algorithms
Extremes of the internal energy of the Potts model on cubic graphs
We prove tight upper and lower bounds on the internal energy per particle
(expected number of monochromatic edges per vertex) in the anti-ferromagnetic
Potts model on cubic graphs at every temperature and for all . This
immediately implies corresponding tight bounds on the anti-ferromagnetic Potts
partition function.
Taking the zero-temperature limit gives new results in extremal
combinatorics: the number of -colorings of a -regular graph, for any , is maximized by a union of 's. This proves the case of a
conjecture of Galvin and Tetali
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