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.

• The paper establishes that every vertex in a Seymour-tight orientation satisfies equal first and second out-neighbourhoods, characterizing examples like directed cycles and regular tournaments.
• It introduces lexicographic and generalized product constructions that scale and preserve tightness, enabling recursive and extremal digraph formations.
• The work provides an algebraic classification for abelian Cayley Seymour-tight orientations, linking additive combinatorics with digraph theory and suggesting directions for non-abelian extensions.

Seymour-tight Orientations: Structure, Constructions, and Algebraic Classification

Introduction and Context

This paper systematically explores "Seymour-tight orientations," a class of oriented graphs motivated by the equality case ($|\NGi{1}{G}{v}| = |\NGi{2}{G}{v}|$ for all $v$) in Seymour's Second Neighbourhood Conjecture (SNC). SNC posits that every oriented graph contains a vertex $v$ for which $|\NGi{2}{G}{v}| \geq |\NGi{1}{G}{v}|$. Absence of such a vertex constitutes a counterexample. While SNC remains unresolved, the very tightness condition (equal first and second out-neighbourhoods everywhere) prompts structural investigation both for its own sake and for its utility in producing "almost counterexamples" and understanding the extremal regime for SNC.

Fundamental Properties and Examples

Seymour-tight orientations are precisely those for which every vertex $v$ has $|\NGi{1}{G}{v}| = |\NGi{2}{G}{v}|$. Canonical examples include directed cycles and regular tournaments. The paper provides precise characterizations: all directed cycles and their $k$th powers (with $n > 2k$) are Seymour-tight. Among tournaments, these are exactly the regular tournaments—a property established via both degree counting and diameter constraints.

The authors extend these examples via structural decompositions: the condensation of any Seymour-tight orientation is a directed acyclic graph whose sinks are themselves Seymour-tight. This gives recursive control over the class.

Constructions: Lexicographic and Generalized Products

A core contribution lies in establishing powerful composition techniques:

• Lexicographic products: If $D$ and $G$ are Seymour-tight, so is $v$0. The construction preserves both the tightness property and, under strong connectivity assumptions, the connectivity structure.
• Generalized lexicographic products: Allowing replacement of each vertex in a host graph $v$1 by a potentially distinct Seymour-tight orientation, yielding a rich and non-regular class of strongly connected Seymour-tight orientations.

The machinery is leveraged to demonstrate that every orientation is an induced subgraph of a strongly connected Seymour-tight orientation, and—if SNC fails—every orientation can be embedded as an induced subgraph into a counterexample. The lexicographic product serves crucially for scaling up putative counterexamples and transferring extremal local behaviours to larger graphs.

Near-counterexamples and Quantitative Results

The lexicographic product framework enables several significant implications:

• If SNC is false, counterexamples exist that are arbitrarily close to regular tournaments in terms of degree (minimum out-degree at least $v$2 for arbitrarily large $v$3, some fixed $v$4).
• Bootstrapping counterexamples via iterated lexicographic products yields orientations of arbitrarily high minimum out-degree violating the conjectured property up to a universal multiplicative constraint: for some $v$5, one obtains $v$6 universally.

Structural and Algebraic Classification

A substantial theoretical advance is provided for Cayley digraphs. Using critical pair theory from additive combinatorics (Kemperman’s theorem) and Hamidoune’s classical result (no Cayley graph is a counterexample to SNC), the authors derive a full structural classification of Seymour-tight orientations that are Cayley digraphs of abelian groups:

• These arise as lexicographic products of empty digraphs, $v$7th powers of directed cycles, and regular tournaments.

This combinatorial–algebraic bridge implies that all abelian Cayley Seymour-tight orientations are fundamentally constructed from these basic components. The paper relates these connection set conditions to additive critical sets, with explicit use of Kemperman's structure theorem.

Non-abelian and Vertex-Transitive Extensions

While the complete classification for non-abelian groups is not settled, the extension to general Cayley digraphs is framed as a central open problem. The authors delineate a route via DeVos’s work—specifically, using trios of sets and appropriate criticality—to potentially extend the classification beyond the abelian case. For vertex-transitive digraphs, the conjectured extension is noted but remains open.

Converse Invariance and Further Structural Questions

An intriguing inquiry concerns converse invariance: under what conditions does the edge-reversal of a Seymour-tight orientation remain Seymour-tight? The equivalence for vertex-transitive cases is established, and a conjecture for the regular case ($v$8-regular, first and second out-neighbourhood) is posited. The authors provide counterexamples in the non-symmetric case using explicit constructions.

Extension to Sullivan’s Conjecture and Related Tightness Notions

Parallel analysis is performed for Sullivan-tight orientations ($v$9 for all $v$0), i.e., those tight for Sullivan's conjecture—a structural analog with in- and out-degree swapped in the SNC premise. The same lexicographic and generalized products preserve Sullivan-tightness, and the structural toolkit is extended to build special putative counterexamples to this related conjecture.

Implications and Future Directions

The theoretical implications are multifold:

• Structural insight: Identification, construction, and classification results for Seymour-tight orientations provide a concrete extreme family close to hypothetical counterexamples. This sharpens understanding of what extremal configurations for SNC must look like.
• Absence of Cayley counterexamples: The full algebraic characterization in the abelian case confirms and extends the connection between additive combinatorics and extremal digraph theory.
• Transferability: Via the lexicographic and generalized products, extremal properties can be compounded, and structural properties of small digraph blocks propagate to large, strongly connected host graphs.
• Future research: Questions concerning converse invariance, classification for more general group actions (non-abelian Cayley or vertex-transitive orientations), and further study of distance-transitive digraphs as subsets of Seymour-tight orientations remain open. Advances in these areas may have direct consequences for progress on SNC itself and for the theory of extremal digraphs.

Conclusion

This paper presents a comprehensive study of Seymour-tight orientations, developing both structural theory and algebraic characterization. The results offer not only new classes of extremal digraphs but also general frameworks for constructing and analyzing orientations near the boundary of SNC. The integration of combinatorial, algebraic, and probabilistic methods points to a rich intersection where progress on open conjectures in digraph theory may be achieved.