2,295 research outputs found
Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult
We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem
Sidorenko's conjecture, colorings and independent sets
Let denote the number of homomorphisms from a graph to a
graph . Sidorenko's conjecture asserts that for any bipartite graph , and
a graph we have where
and denote the number of vertices and edges of the graph and
, respectively. In this paper we prove Sidorenko's conjecture for certain
special graphs : for the complete graph on vertices, for a
with a loop added at one of the end vertices, and for a path on vertices
with a loop added at each vertex. These cases correspond to counting colorings,
independent sets and Widom-Rowlinson colorings of a graph . For instance,
for a bipartite graph the number of -colorings
satisfies
In fact, we will prove that in the last two cases (independent sets and
Widom-Rowlinson colorings) the graph does not need to be bipartite. In all
cases, we first prove a certain correlation inequality which implies
Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande
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