## Physical Sciences Oral Presentations

#### Title

Uniquely and 2-Uniquely Hamiltonian Graphs

Presentation

Mark Kayll

Mathematics

#### Abstract / Artist's Statement

In graph theory a Hamilton cycle is a walk around the vertices of a graph in which each vertex is visited exactly once, and then it returns to the starting vertex. The problem of determining whether a graph contains a Hamilton cycle has been studied extensively and is determined to belong to the so-called NP-complete family of problems for arbitrary graphs. Due to the difficulty in solving such a problem for an arbitrary graph, we set our sights on a family of graphs described by graph theorist John Sheehan. A maximum uniquely Hamiltonian graph contains the greatest number of edges possible while maintaining a single Hamilton cycle. Sheehan shows that for a graph with n nodes (for n >= 4), the maximum number of edges it can contain is equal to (n^2/4) + 1. We will describe an algorithm that finds the Hamilton cycle for any such graph or any of its subgraphs in polynomial time. This algorithm shows that these graphs do not suffer the same complexity issues as do arbitrary graphs for a Hamilton cycle problem. For any graph containing a single Hamilton cycle, that cycle can be revealed in polynomial time.

#### Category

Physical Sciences

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Uniquely and 2-Uniquely Hamiltonian Graphs

In graph theory a Hamilton cycle is a walk around the vertices of a graph in which each vertex is visited exactly once, and then it returns to the starting vertex. The problem of determining whether a graph contains a Hamilton cycle has been studied extensively and is determined to belong to the so-called NP-complete family of problems for arbitrary graphs. Due to the difficulty in solving such a problem for an arbitrary graph, we set our sights on a family of graphs described by graph theorist John Sheehan. A maximum uniquely Hamiltonian graph contains the greatest number of edges possible while maintaining a single Hamilton cycle. Sheehan shows that for a graph with n nodes (for n >= 4), the maximum number of edges it can contain is equal to (n^2/4) + 1. We will describe an algorithm that finds the Hamilton cycle for any such graph or any of its subgraphs in polynomial time. This algorithm shows that these graphs do not suffer the same complexity issues as do arbitrary graphs for a Hamilton cycle problem. For any graph containing a single Hamilton cycle, that cycle can be revealed in polynomial time.