Variations on Sidorenko's conjecture in tournaments

Abstract

We study variants of Sidorenko's conjecture in tournaments, where new phenomena arise that do not have clear analogues in the setting of undirected graphs. We first consider oriented graphs that are systematically under-represented in tournaments (called tournament anti-Sidorenko). We prove that such oriented graphs must be quite sparse; specifically, the maximum number of edges of a $k$-vertex oriented graph which is tournament anti-Sidorenko is $(1+o(1))k\log_2 k$. We also give several novel constructions of oriented graphs that are systematically over-represented in tournaments (tournament Sidorenko); as a representative example, we show that most ways to delete an edge from a transitive tournament yield a tournament Sidorenko oriented graph. As an illustration of our methods, we characterize which orientations of stars are tournament Sidorenko and which are tournament anti-Sidorenko.

Figures (2)

• Figure 1: The above four vertex graph has the tournament Sidorenko property, since the induced digraph on $\{a, b, c\}$ has the directed Sidorenko property and $v$ is complete to $\{a,b,c\}$. This is a special case of \ref{['c:growspecial']}.
• Figure 2: Per \ref{['p:even-star']}, we have that $(I_1, D_1)$ is strongly tournament anti-Sidorenko with $I_1 = \{w\}$, and per \ref{['p:anti2']}$(I_2, D_2)$ is strongly tournament anti-Sidorenko with $I_2 = \{a, b\}$. Then by \ref{['l:glue-anti-sid']} the digraph $D$ formed by identifying $v \in D_1$ with $w \in D_2$ satisfies $(I_2, D)$ is strongly tournament anti-Sidorenko.

Theorems & Definitions (82)

• Definition 1.1
• Conjecture 1.2: Sidorenko's Conjecture
• Definition 1.3
• Remark 1.4
• Definition 1.5: Tournament Sidorenko properties
• Remark 1.6
• Theorem 1.7
• Definition 1.8
• Proposition 1.9
• Theorem 1.10