Event Title

Maximum Sk packing for λ-fold complete graph

Faculty Advisor

Daniel Roberts

Graduation Year

2020

Location

Room E102, Center for Natural Sciences, Illinois Wesleyan University

Start Date

4-4-2020 11:15 AM

End Date

4-4-2020 11:30 AM

Description

A complete graph Kn is defined as a simple graph in which every vertex is connected to each other by exactly one edge. A k-star, denoted by Sk, is a graph on k + 1 vertices with exactly one vertex of degree k and all other vertices of degree 1. An Sk-decomposition of a Kn is a partition of the edge set of Kn where each block of the partition is isomorphic to Sk. A λ-fold complete graph is a complete graph where each edge is repeated λ times. The necessary and sufficient conditions for an Sk-decomposition of Kn are known. Particularly, we will study the cases where such a decomposition does not exist and characterize the number of copies of Sk that can fit into a regular complete graph (λ = 1) and a 2-fold complete graph (λ=2). We also intend to investigate the structure of the leftover edges.

In addition, we found possible general cases of leave cardinalities and some leave graphs of the leave of a maximum packing of Kn with k-stars by using Java and Python as the baseline of the experiment. We output CSV files of all possible leave cardinalities for Sk-packings of Kn and found leave graphs assisting with the computer programming language, which can shorten the construction time and make it easier to use switching argument.

This document is currently not available here.

Share

COinS
 
Apr 4th, 11:15 AM Apr 4th, 11:30 AM

Maximum Sk packing for λ-fold complete graph

Room E102, Center for Natural Sciences, Illinois Wesleyan University

A complete graph Kn is defined as a simple graph in which every vertex is connected to each other by exactly one edge. A k-star, denoted by Sk, is a graph on k + 1 vertices with exactly one vertex of degree k and all other vertices of degree 1. An Sk-decomposition of a Kn is a partition of the edge set of Kn where each block of the partition is isomorphic to Sk. A λ-fold complete graph is a complete graph where each edge is repeated λ times. The necessary and sufficient conditions for an Sk-decomposition of Kn are known. Particularly, we will study the cases where such a decomposition does not exist and characterize the number of copies of Sk that can fit into a regular complete graph (λ = 1) and a 2-fold complete graph (λ=2). We also intend to investigate the structure of the leftover edges.

In addition, we found possible general cases of leave cardinalities and some leave graphs of the leave of a maximum packing of Kn with k-stars by using Java and Python as the baseline of the experiment. We output CSV files of all possible leave cardinalities for Sk-packings of Kn and found leave graphs assisting with the computer programming language, which can shorten the construction time and make it easier to use switching argument.