Abstract:
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We consider the H-coloring problem on graphs with
vertices of large degree. We prove that for H an odd cycle,
the problem belongs to P. We also study the phase transition
of the problem, for an infinite family of graphs of a given
chromatic number, i.e. the threshold density value for which
the problem changes from P to NP-complete. We extend the result
for the case that the input graph has a logarithmic size of
small degree vertices.As a corollary, we get a new result on
the chromatic number; a new family of graphs, for which computing
the chromatic number can be done in polynomial time. |