# A reduction of the spectrum problem for odd sun systems and the prime case

## TL;DR

This paper extends the existence results for $k$-sun decompositions of complete graphs with odd $k$, showing that if such decompositions exist within a specific range, they exist for all larger admissible values.

## Contribution

It adapts a method from odd cycle systems to reduce the spectrum problem for odd sun systems, establishing a broad existence criterion.

## Key findings

- Existence of $k$-sun systems for all admissible $v \, \geq 6k$ under certain conditions.
- Reduction of the spectrum problem to a finite range for odd sun systems.
- Extension of known results from small cases to all larger admissible cases.

## Abstract

A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun. The existence problem for $k$-sun decompositions of $K_v$, with $k$ odd, has been solved only when $k=3$ or $5$. By adapting a method used by Hoffmann, Lindner and Rodger to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a $k$-sun system of $K_v$ ($k$ odd) whenever $v$ lies in the range $2k< v < 6k$ and satisfies the obvious necessary conditions, then such a system exists for every admissible $v\geq 6k$.