## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, December 03, 2013, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Choongbum Lee (MIT)

## Two approaches to Sidorenko's conjecture

Sidorenko's conjecture states that the number of homomorphisms from a bipartite graph H to a graph G is at least the expected number of homomorphisms from H to the binomial random graph with the same expected edge density as G. In this talk, I will present two approaches to the conjecture. First, I will introduce the notion of tree-arrangeability, where a bipartite graph H with bipartition A \cup B is tree-arrangeable if neighborhoods of vertices in A have a certain tree-like structure, and show that Sidorenko's conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's conjecture holds if there are two vertices a1, a2 in A such that each vertex a in A satisfies N(a) \subseteq N(a1) or N(a) \subseteq N(a_2). Second, I will prove that if T is a tree and H is a bipartite graph satisfying Sidorenko's conjecture, then the Cartesian product of T and H also satisfies Sidorenko's conjecture. This result implies that, for all d \ge 2, the d-dimensional grid with arbitrary side lengths satisfies Sidorenko's conjecture.

Joint with Jeong Han Kim (KIAS) and Joonkyung Lee (Oxford)

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