Uniquely K_r-Saturated Graphs

A graph G is uniquely K_r-saturated if it contains no clique with r vertices and if for all edges e in the complement, G + e has a unique clique with r vertices. Previously, few examples of uniquely K_r-saturated graphs were known, and little was known about their properties. We search for these graphs by adapting orbital branching, a technique originally developed for symmetric integer linear programs. We find several new uniquely K_r-saturated graphs with 4 \leq r \leq 7, as well as two new infinite families based on Cayley graphs for Z_n with a small number of generators.

💡 Research Summary

The paper investigates a specialized class of graphs known as uniquely Kₙ‑saturated graphs. A graph G is uniquely Kᵣ‑saturated if it contains no r‑clique, yet for every non‑edge e in the complement (\overline{G}) the graph G + e contains exactly one r‑clique. When such a graph also lacks a dominating vertex it is called r‑primitive. Prior work had only a handful of examples (mainly for r = 3) and very little understanding of their structure.

To address this gap the authors adapt a technique from symmetric integer programming called orbital branching. They model the construction of a graph as a trigraph, where each unordered pair of vertices is colored black (edge), white (non‑edge) or gray (undecided). Two fundamental constraints are imposed on any partial trigraph: (C1) it must not already contain a black r‑clique, and (C2) each vertex pair may have at most one black Kᵣ‑completion. A third, stronger constraint (C3) is added during the search: whenever a pair is set to white, a unique set of r‑2 vertices must be selected that together with the pair form a Kᵣ‑completion, and all required edges are forced black.

Orbital branching exploits the automorphism group of the current trigraph. An orbit of gray pairs is selected; one branch colors a single representative white (and then applies the custom augmentation to enforce (C3)), while the other branch colors the entire orbit black. The custom augmentation dramatically reduces the search space because it forces many edges at once and eliminates many symmetric duplicates by considering orbits of the (r‑2)‑subsets under the stabilizer of the chosen white pair.

Using this algorithm the authors performed exhaustive searches for r = 4, 5, 6, 7 up to certain vertex limits (N₄ = 20, N₅ = N₆ = 16, N₇ = 17). They discovered ten new r‑primitive graphs, most of which have no simple description and are labeled G(N)ₐ, G(N)ᵦ, etc. Two of the new graphs are vertex‑transitive with a prime number of vertices, implying they are Cayley graphs. One is the Paley graph of order 13 (a 4‑primitive graph); the other is the complement of the Cayley graph C(ℤ₁₇,{1,4}), which is a 7‑primitive graph.

Motivated by the Cayley example, the authors systematically searched for r‑primitive graphs among complements of Cayley graphs on ℤₙ with a small generating set S. By enumerating generator sets of size two and three, they observed regular patterns and proved two infinite families:

• Theorem 2: For t ≥ 2, let n = 4t² + 1 and r = 2t² − t + 1. Then the complement of the Cayley graph C(ℤₙ,{1,2t}) is r‑primitive.
• Theorem 3: For t ≥ 2, let n = 9t² − 3t + 1 and r = 3t² − 2t + 1. Then the complement of the Cayley graph C(ℤₙ,{1,3t − 1,3t}) is r‑primitive.

The proofs rely on exact computation of the clique number for these circulant graphs and a discharging argument for the three‑generator family. The authors note that computing clique numbers for Cayley or circulant graphs is notoriously hard, and their constructive approach provides new insight.

A striking by‑product is a counterexample to the previously observed regularity of r‑primitive graphs: a 5‑primitive graph on 16 vertices with minimum degree 8 and maximum degree 9, showing that regularity is not mandatory.

Finally, the paper formulates Conjecture 1: for each r ≥ 3 there are only finitely many r‑primitive graphs. The conjecture is known only for r = 3 (the Moore graphs) and remains open for larger r. The authors discuss open directions, including the existence of infinite families of Cayley complements with four or more generators, the role of roots of unity in generator sets, and the need for bounds on clique and independence densities in infinite Cayley graphs, which could improve algorithmic searches.

Overall, the work combines sophisticated symmetry‑exploiting algorithms with algebraic constructions to substantially enlarge the catalog of uniquely Kᵣ‑saturated graphs, provides the first infinite families beyond the trivial odd‑cycle complements, and raises several compelling structural questions for future research.