Leaves for packings with block size four

TL;DR

This paper studies maximum packings of 4-cliques in complete graphs, focusing on the structure of uncovered edges (leaves), especially 2-regular leaves, and provides constructions and bounds for various cycle lengths.

Contribution

It extends previous work on leaves in maximum packings, offering new constructions and bounds for 2-regular leaves with specified cycle lengths.

Findings

Extended constructions for 2-regular leaves.

Established lower bounds for existence of packings with given leaves.

Analyzed leaves with cycle lengths in specified subsets.

Abstract

We consider maximum packings of edge-disjoint $4$-cliques in the complete graph $K_n$. When $n \equiv 1$ or $4 \pmod{12}$, these are simply block designs. In other congruence classes, there are necessarily uncovered edges; we examine the possible `leave' graphs induced by those edges. We give particular emphasis to the case $n \equiv 0$ or $3 \pmod{12}$, when the leave is $2$-regular. Colbourn and Ling settled the case of Hamiltonian leaves in this case. We extend their construction and use several additional direct and recursive constructions to realize a variety of $2$-regular leaves. For various subsets $S \subseteq \{3,4,5,\dots\}$, we establish explicit lower bounds on $n$ to guarantee the existence of maximum packings with any possible leave whose cycle lengths belong to $S$.