# The Doyen-Wilson theorem for 3-sun systems

## TL;DR

This paper provides a complete solution to the existence problem of G-designs with G being a 3-sun, extending previous results for triangles, kite, and bull designs in combinatorial design theory.

## Contribution

It extends the Doyen-Wilson theorem to 3-sun systems, solving the existence problem for these specific G-designs.

## Key findings

- Complete characterization of 3-sun G-design existence
- Extension of Doyen-Wilson theorem to new graph class
- Resolution of a longstanding combinatorial design problem

## Abstract

A solution to the existence problem of G-designs with given subdesigns is known when G is a triangle with p=0,1, or 2 disjoint pendent edges: for p=0, it is due to Doyen and Wilson, the first to pose such a problem for Steiner triple systems; for p=1 and p=2, the corresponding designs are kite systems and bull designs, respectively. Here, a complete solution to the problem is given in the remaining case where G is a 3-sun, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor.