Uniquely K_r-Saturated Graphs

TL;DR

This paper explores the properties and construction of uniquely K_r-saturated graphs, introducing new examples and infinite families using orbital branching and Cayley graphs.

Contribution

It introduces a novel application of orbital branching to find new uniquely K_r-saturated graphs and identifies two new infinite families based on Cayley graphs.

Findings

Discovered several new uniquely K_r-saturated graphs for 4 ≤ r ≤ 7

Identified two infinite families of such graphs based on Cayley graphs

Demonstrated the effectiveness of orbital branching in graph discovery

Abstract

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.