Design spectra for 6-regular graphs with 12 vertices

TL;DR

This work determines the design spectra for twelve-vertex, six-regular graphs by combining Wilson’s existence framework with group divisible designs to construct edgewise decompositions of complete graphs into copies of a fixed 6-vertex, 12-vertex graph. The authors show that for a large class of graphs the design order must satisfy $n \equiv 1 \pmod{72}$, and they provide both direct (for specific graphs and orders) and general (GDD-based) constructions to cover thousands of graphs. Their main result asserts that, for $7788$ such graphs, a $G$-design of order $n$ exists if and only if $n \equiv 1 \pmod{72}$, with detailed appendix data and several well-chosen exceptional cases. The work highlights the power of group divisible designs in resolving design spectra for dense regular graphs and discusses where scarcity of certain GDD types (notably 5-GDDs) limits immediate reach, pointing to avenues for future breakthroughs.

Abstract

The design spectrum of a simple graph $G$ is the set of positive integers $n$ such that there exists an edgewise decomposition of the complete graph $K_n$ into $n(n - 1)/(2 |E(G)|)$ copies of $G$. We compute the design spectra for 7788 6-regular graphs with 12 vertices.

Figures (3)

• Figure 1: Vertex-transitive graphs
• Figure 2: Vertex-transitive graphs
• Figure 3: Graphs

Theorems & Definitions (28)

• Lemma 3.1: AbelColbournDinitz2007AbelGeGreigLing2009BrouwerSchrijverHanani1977Ge2007GeLing2004cGeLing2005GeMiao2007HananiRayChaudhuriWilson1972WeiGe2014
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof