Seymour-tight orientations

Abstract

We investigate `almost counterexamples' to Seymour's second neighbourhood conjecture. In what we call Seymour-tight orientations, the size of the first neighbourhood of each vertex equals the size of its second neighbourhood. We give several examples and constructions. Specifically, we prove that the class of Seymour-tight orientations is closed under taking (generalized) lexicographic products. Moreover, the lexicographic product of a putative counterexample to Seymour's second neighbourhood conjecture and a Seymour-tight orientation is again a counterexample. Using lexicographic products, we show that if the conjecture is false, then there exist counterexamples that are close to regular tournaments, and moreover that any digraph occurs as an induced subgraph of a counterexample. We then use this same machinery to construct special putative counterexamples to Sullivan's conjecture. The inherent symmetry of these orientations give access to an algebraic perspective. Seymour-tight orientations that are also Cayley digraphs correspond to special pairs of critical sets in groups, which connects potentially to additive combinatorics. We use Kemperman's theorem to characterize those Seymour-tight orientations that are the Cayley digraph of an abelian group.

Figures (5)

• Figure 1: Four different ways to construct a larger Seymour-tight orientation from $\mkern2mu\overrightarrow{\mkern-2mu{C_3}\newline}\mkern2mu$. (1) Take a lexicographic product $\mkern2mu\overrightarrow{\mkern-2mu{C_3}\newline}\mkern2mu[E_2]$ (Lemma \ref{['Seymour lex']}). (2) Take a generalized lexicographic product $\mkern2mu\overrightarrow{\mkern-2mu{C_3}\newline}\mkern2mu[\mkern2mu\overrightarrow{\mkern-2mu{C_3}\newline}\mkern2mu,E_3,E_3]$ (Corollary \ref{['vervang alle punten']}). (3) Add a source $s$ such that $N^{}_{1}(G,s)= N^{}_{1}(G,v)$ for some $v \in V(G)$ (Lemma \ref{['Source copy neighbourhood']}). (4) Use a digraph homomorphism $G \rightarrow H$ to add a source component $G$ to $H$ (Lemma \ref{['Graph hom construction']}). Note that (1) and (2) are strongly connected, whereas (3) and (4) are not.
• Figure 2: Two examples of a strongly connected non-regular Seymour-tight orientations.
• Figure 3: A generalized lexicographic product $D[G_1,\ldots,G_6]$ where $|V(G_1)|=|V(G_3)|=|V(G_5)|=3$ and $|V(G_2)=|V(G_4)|=|V(G_6)|=1$.
• Figure 4: If a large regular tournament on $2k+1$ vertices contains a small regular tournament such that all other vertices are uniform on that tournament, then we can replace the small regular tournament with some other Seymour-tight orientation.
• Figure 5: Three Sullivan-tight orientations that are not Seymour-tight.

Theorems & Definitions (68)

• Conjecture 1.1: Seymour dean1995squaring
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2: Theorem 4.2 brantner2009contributions