Randomness friendly graphs

Abstract

We consider the two problems from extremal graph theory: 1. Given integer N, real p ϵ (0, 1) and a graph G, what is the minimum number of copies of G a graph H with N vertices and pN2/2 edges can contain? 2. Given an integer N and a graph G, what is the minimum number of copies of G an N‐vertex graph H and its complement H¯ can contain altogether? In each of the problems, we say that G is “randomness friendly” if the number of its copies is nearly minimal when H is the random graph. We investigate how the two classes of graphs are related: the graphs which are “randomness friendly” in Problem 1 and those of Problem 2. In the latter problem, we discover new families of graphs which are “randomness friendly.” © 1996 John Wiley & Sons, Inc.

Non-Three-Colourable Common Graphs Exist

“…We will prove in Theorem 3.1 that W 5 (see Figure 1) is common. This will also answer a question of Sidorenko [19]. He showed [19,Theorem 8] that every graph that is obtained by adding a vertex of full degree to a bipartite graph of average degree at least one satisfying the Erdős-Simonovits-Sidorenko conjecture is common.…”

“…This will also answer a question of Sidorenko [19]. He showed [19,Theorem 8] that every graph that is obtained by adding a vertex of full degree to a bipartite graph of average degree at least one satisfying the Erdős-Simonovits-Sidorenko conjecture is common. Sidorenko further asked whether, in this theorem, both conditions of being bipartite and having average degree at least one are essential in order to obtain a common graph.…”

On tripartite common graphs

“…By Sidorenko's theorem [27], the 4-wheel K 1,2,2 is common; however, m C 4 (W) = 1/8 if and only if W = 1/2 almost everywhere, i.e., W is quasirandom, the naive approach using commonality of K 1,2,2 while bounding m C 4 from above does not work. We circumvent this difficulty by comparing m K 1,2,2 (W) and m C 4 (W).…”

Locally common graphs

“…, which can be interpreted as the probability that a random map ϕ V F V G : ( ) ( ) → preserves adjacency. With this notation, common graphs are those graphs F for which [18] studied various "convexity" properties of graphs, one of which is closely related to common graphs. Let us say that a graph F has the Sidorenko property if for every graph G,…”