5,140 research outputs found
Ramsey multiplicity and the Tur\'an coloring
Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that
among all two-colorings of the edges of a complete graph, the uniformly random
coloring asymptotically minimizes the number of monochromatic copies of any
fixed graph . This conjecture was disproved independently by Sidorenko and
Thomason. The first author later found quantitatively stronger counterexamples,
using the Tur\'an coloring, in which one of the two colors spans a balanced
complete multipartite graph.
We prove that the Tur\'an coloring is extremal for an infinite family of
graphs, and that it is the unique extremal coloring.
This yields the first determination of the Ramsey multiplicity constant of a
graph for which the Burr--Rosta conjecture fails.
We also prove an analogous three-color result. In this case, our result is
conditional on a certain natural conjecture on the behavior of two-color Ramsey
numbers.Comment: 37 page
Improved Inapproximability Results for Maximum k-Colorable Subgraph
We study the maximization version of the fundamental graph coloring problem.
Here the goal is to color the vertices of a k-colorable graph with k colors so
that a maximum fraction of edges are properly colored (i.e. their endpoints
receive different colors). A random k-coloring properly colors an expected
fraction 1-1/k of edges. We prove that given a graph promised to be
k-colorable, it is NP-hard to find a k-coloring that properly colors more than
a fraction ~1-O(1/k} of edges. Previously, only a hardness factor of 1-O(1/k^2)
was known. Our result pins down the correct asymptotic dependence of the
approximation factor on k. Along the way, we prove that approximating the
Maximum 3-colorable subgraph problem within a factor greater than 32/33 is
NP-hard. Using semidefinite programming, it is known that one can do better
than a random coloring and properly color a fraction 1-1/k +2 ln k/k^2 of edges
in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to
properly color (using k colors) more than a fraction 1-1/k + O(ln k/ k^2) of
edges of a k-colorable graph.Comment: 16 pages, 2 figure
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