On graph decompositions: Exploring the stars and stripes problem with odd n-stars

Author

Jehyun Lee, Michigan Technological UniversityFollow

Date of Award

2025

Document Type

Campus Access Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Melissa S. Keranen

Committee Member 1

William J. Keith

Committee Member 2

Robert P. Schneider

Committee Member 3

Soner Onder

Abstract

This dissertation addresses the “Stars and Stripes" problem, which is a graph decomposition problem. The Stars and Stripes problem can be described as finding uniformly resolvable decompositions when the blocks are either $n$-stars or edges.

In other words, this refers to decomposing a complete graph into spanning sub-graphs, these spanning sub-graphs are either a union of disjoint $n$-stars or a union of disjoint edges. We will call these $n$-star factors and $1$-factors respectively. A given complete graph which satisfies the necessary condition (\ref{ness}) can potentially be decomposed into $r$ $1$-factors and $s$ $n$-star factors. The goal of this problem is to find a solution for every pair $(r,s)$ and every complete graph that satisfies the necessary conditions. This dissertation focuses on the odd $n$-star cases of the problem. We present the solutions we have found along with the remaining open cases.

In Chapter \ref{ch2 WC}, we present a method for decomposing a complete graph into cycles using weighted graphs. This method provides solutions for most cases where $v$ is large. However, this method requires our decomposition to contain a certain number of $1$-factors. Thus, we cannot find solutions for extreme cases, that is $(r,s)$ pairs when $r$ is very small.

Because the technique in the Chapter \ref{ch2 WC} does not apply to extreme cases, we began to study the most extreme cases, where $r=1$. In Chapter \ref{ch3 5star}, we give solutions for the case when $r=1$ and $n=5$. The generalized version, when $r=1$ and $n$ is odd, is presented in Chapter \ref{ch4 Nstar}. The results in Chapters \ref{ch3 5star} and \ref{ch4 Nstar} are obtained fundamentally by the same method. However, the constructions in Chapter \ref{ch4 Nstar} are much more complex, and there are many more detailed cases.

Recommended Citation

Lee, Jehyun, "On graph decompositions: Exploring the stars and stripes problem with odd n-stars", Campus Access Dissertation, Michigan Technological University, 2025.

https://doi.org/10.37099/mtu.dc.etdr/1940