The Bipartite Swapping Trick on Graph Homomorphisms

Details The Bipartite Swapping Trick on Graph Homomorphisms The Bipartite Swapping Trick on Graph Homomorphisms

2011-04-19

arxiv.org/abs/1104.3704v1

Mathematics

Combinatorics

22 pages. To appear in SIAM J. Discrete Math

Scientific paper

We provide an upper bound to the number of graph homomorphisms from $G$ to $H$, where $H$ is a fixed graph with certain properties, and $G$ varies over all $N$-vertex, $d$-regular graphs. This result generalizes a recently resolved conjecture of Alon and Kahn on the number of independent sets. We build on the work of Galvin and Tetali, who studied the number of graph homomorphisms from $G$ to $H$ when $H$ is bipartite. We also apply our techniques to graph colorings and stable set polytopes.

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