Uniformly resolvable decompositions of $K_v$ into $1$-factors and odd $n$-star factors

TL;DR

The paper addresses uniformly resolvable decompositions of $K_v$ into $1$-factors and odd $n$-star factors by focusing on $(K_2,K_{1,n})$-URDs with odd $n$. It develops a weighted-graph framework and almost uniformly resolvable decompositions (AURD) to combine cycle decompositions of $K_{m(n+1)}$ with the multipartite edges $K_{n+1}^x$, enabling a staged construction of URDs. Necessary conditions are established: there exists an integer $x$ with $0 \le x \le \left\lfloor \frac{v-1}{2n} \right\rfloor$ such that $s=(n+1)x$ and $r=v-1-2nx$, with parity and divisibility requirements on $v$, and all constructions are built around these relations. The main contribution is a set of explicit parameter families for which $(K_2,K_{1,n})$-URD$(K_v; r,s)$ exists, together with detailed combinatorial schemes to fill inner edges, yielding a substantial partial solution to the existence problem for uniformly resolvable decompositions into $1$-factors and odd $n$-star factors.

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 partial solution for the case in which all resolution classes are either $K_2$ or $K_{1,n}$ where $n$ is odd.

Figures (1)

• Figure 1: A cycle $C_m$ and a weighted cycle $C_{m(n+1)}$

Theorems & Definitions (17)

• Lemma 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• Lemma 4.1