#### 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 K_{n} is defined as a simple graph in which every vertex is connected to each other by exactly one edge. A k-star, denoted by S_{k}, is a graph on k + 1 vertices with exactly one vertex of degree k and all other vertices of degree 1. An S_{k}-decomposition of a K_{n} is a partition of the edge set of Kn where each block of the partition is isomorphic to S_{k}. A λ-fold complete graph is a complete graph where each edge is repeated λ times. The necessary and sufficient conditions for an S_{k}-decomposition of K_{n} are known. Particularly, we will study the cases where such a decomposition does not exist and characterize the number of copies of S_{k} 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 K_{n} with k-stars by using Java and Python as the baseline of the experiment. We output CSV files of all possible leave cardinalities for S_{k}-packings of K_{n} and found leave graphs assisting with the computer programming language, which can shorten the construction time and make it easier to use switching argument.

Maximum Sk packing for λ-fold complete graph

Room E102, Center for Natural Sciences, Illinois Wesleyan University

A complete graph K_{n} is defined as a simple graph in which every vertex is connected to each other by exactly one edge. A k-star, denoted by S_{k}, is a graph on k + 1 vertices with exactly one vertex of degree k and all other vertices of degree 1. An S_{k}-decomposition of a K_{n} is a partition of the edge set of Kn where each block of the partition is isomorphic to S_{k}. A λ-fold complete graph is a complete graph where each edge is repeated λ times. The necessary and sufficient conditions for an S_{k}-decomposition of K_{n} are known. Particularly, we will study the cases where such a decomposition does not exist and characterize the number of copies of S_{k} 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 K_{n} with k-stars by using Java and Python as the baseline of the experiment. We output CSV files of all possible leave cardinalities for S_{k}-packings of K_{n} and found leave graphs assisting with the computer programming language, which can shorten the construction time and make it easier to use switching argument.