Uniformly resolvable decompositions of $K_v$ into one $1$-factor and $n$-stars when $n>1$ is odd

TL;DR

This work resolves the existence of uniformly resolvable decompositions of $K_v$ into one $1$-factor and $n$-stars with odd $n>1$, proving that such a decomposition exists precisely when $v \equiv 2(n+1) \pmod{n(n+1)}$. The authors develop a multi-stage construction: first building almost $n$-star factors on a smaller graph, then lifting them to $K_v$ via Part I and Part II factors, and finally coordinating the remaining edges with Balanced Star Arrays. The key contributions include a complete characterization for the $(K_2,K_{1,n})$-URD problem under the stated constraint, and a concrete, modular construction framework that uses pure, prime, mixed, and little-star components to control edge differences. This advances the theory of uniformly resolvable decompositions and provides a constructive method potentially extensible to other target graphs.

Abstract

We consider uniformly resolvable decompositions of $K_v$ into subgraphs such that each resolution class contains only blocks isomorphic to the same graph. We give a complete solution for the case in which one resolution class is $K_2$ and the rest are $K_{1,n}$ where $n>1$ is odd.

Figures (3)

• Figure 1: An almost $5$-star factor on $g=27$ vertices
• Figure 2: Illustration of a partial $5$-star factor on $v=162$ vertices from the almost $5$-star factor on $27$ vertices in Figure$~\ref{['v27']}$
• Figure 3: Balanced star array for $v=162$

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof