Embedding an Edge-colored $K(a^{(p)};\lambda,\mu )$ into a Hamiltonian Decomposition of $K(a^{(p+r)};\lambda,\mu )$

TL;DR

This paper investigates conditions under which a graph decomposition of a specific multipartite graph can be extended to a Hamiltonian decomposition of a larger, related graph, providing general and specific solutions for various parameters.

Contribution

The paper introduces a general theorem for embedding graph decompositions into Hamiltonian decompositions of larger graphs, solving the problem for all relevant parameter ranges.

Findings

Established a general embedding theorem for Hamiltonian decompositions.

Solved the embedding problem for minimal and specific values of r.

Provided explicit conditions for extending decompositions in multipartite graphs.

Abstract

Let $K(a^{(p)};\lambda,\mu )$ be a graph with $p$ parts, each part having size $a$, in which the multiplicity of each pair of vertices in the same part (in different parts) is $\lambda$ ($\mu $, respectively). In this paper we consider the following embedding problem: When can a graph decomposition of $K(a^{(p)};\lambda,\mu )$ be extended to a Hamiltonian decomposition of $K(a^{(p+r)};\lambda,\mu )$ for $r>0$? A general result is proved, which is then used to solve the embedding problem for all $r\geq \frac{\lambda}{\mu a}+\frac{p-1}{a-1}$. The problem is also solved when $r$ is as small as possible in two different senses, namely when $r=1$ and when $r=\frac{\lambda}{\mu a}-p+1$.