Group divisible (K_4-e)-packings with any minimum leave

TL;DR

This paper investigates the conditions for decomposing complete n-partite graphs minus a minimal leave into edge-disjoint copies of a specific graph, advancing understanding of group divisible packings in combinatorics.

Contribution

It establishes necessary and sufficient conditions for the existence of maximum group divisible $(K_4-e)$-packings with minimal leaves.

Findings

Characterization of all possible minimum leaves

Necessary and sufficient existence conditions

Extension of packing theory to new graph classes

Abstract

A decomposition of $K_{n(g)}\setminus L$, the complete n-partite equipartite graph with a subgraph L (called the leave) removed, into edge disjoint copies of a graph G is called a maximum group divisible packing of $K_{n(g)}$ with G if L contains as few edges as possible. We examine all possible minimum leaves for maximum group divisible $(K_4-e)$-packings. Necessary and sufficient conditions are established for their existences.