## Oral Presentations--Session 6

#### Event Title

Maximum Sk packing for λ-fold complete graph

Daniel Roberts

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.

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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.