Smaller embeddings of partial $k$-star decompositions

TL;DR

This paper investigates conditions under which partial $k$-star decompositions of complete graphs can be embedded into full $k$-star decompositions of larger complete graphs, improving previous bounds on the size increase needed.

Contribution

The authors improve bounds on the minimal size increase required for embedding partial $k$-star decompositions into complete $k$-star decompositions, refining earlier results by Noble and Richardson.

Findings

Any partial $k$-star decomposition of $K_n$ can be embedded in a $k$-star decomposition of $K_{n+s}$ with $s < rac{9}{4}k$ for odd $k$.

For even $k$, the embedding is possible with $s < (6-2 ext{sqrt}(2))k$.

Constants for general $k$ are proven to be optimal and cannot be improved.

Abstract

A $k$-star is a complete bipartite graph $K_{1,k}$. For a graph $G$, a $k$-star decomposition of $G$ is a set of $k$-stars in $G$ whose edge sets partition the edge set of $G$. If we weaken this condition to only demand that each edge of $G$ is in at most one $k$-star, then the resulting object is a partial $k$-star decomposition of $G$. An embedding of a partial $k$-star decomposition $\mathcal{A}$ of a graph $G$ is a partial $k$-star decomposition $\mathcal{B}$ of another graph $H$ such that $\mathcal{A} \subseteq \mathcal{B}$ and $G$ is a subgraph of $H$. This paper considers the problem of when a partial $k$-star decomposition of $K_n$ can be embedded in a $k$-star decomposition of $K_{n+s}$ for a given integer $s$. We improve a result of Noble and Richardson, itself an improvement of a result of Hoffman and Roberts, by showing that any partial $k$-star decomposition of $K_n$ can be embedded in a $k$-star decomposition of $K_{n+s}$ for some $s$ such that $s < \frac{9}{4}k$ when $k$ is odd and $s < (6-2\sqrt{2})k$ when $k$ is even. For general $k$, these constants cannot be improved. We also obtain stronger results subject to placing a lower bound on $n$.