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The Structure of Maximum Independent Sets in Fullerenes Carly Vollet Portland State University carlyw@pdx.edu - p. 1/28

Outline of the talk ■ Introduction/History ● Outline of the talk Introduction/History What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 2/28

Outline of the talk ■ Introduction/History ● Outline of the talk ■ What is a Fullerene? Introduction/History What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 2/28

Outline of the talk ■ Introduction/History ● Outline of the talk ■ What is a Fullerene? Introduction/History What is a fullerene? ■ Coloring and Counting Lemmas Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 2/28

Outline of the talk ■ Introduction/History ● Outline of the talk ■ What is a Fullerene? Introduction/History What is a fullerene? ■ Coloring and Counting Lemmas Counting and Coloring Lemmas ■ Path Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 2/28

Outline of the talk ■ Introduction/History ● Outline of the talk ■ What is a Fullerene? Introduction/History What is a fullerene? ■ Coloring and Counting Lemmas Counting and Coloring Lemmas ■ Path Lemmas Path Lemmas ■ Statement of the Main Result Main Result A tangible result Acknowledgments - p. 2/28

Outline of the talk ■ Introduction/History ● Outline of the talk ■ What is a Fullerene? Introduction/History What is a fullerene? ■ Coloring and Counting Lemmas Counting and Coloring Lemmas ■ Path Lemmas Path Lemmas ■ Statement of the Main Result Main Result ■ A Tangible Result A tangible result Acknowledgments - p. 2/28

Outline of the talk ■ Introduction/History ● Outline of the talk ■ What is a Fullerene? Introduction/History What is a fullerene? ■ Coloring and Counting Lemmas Counting and Coloring Lemmas ■ Path Lemmas Path Lemmas ■ Statement of the Main Result Main Result ■ A Tangible Result A tangible result ■ Acknowledgments Acknowledgments - p. 2/28

History ■ In chemistry, a fullerene refers to a family of carbon ● Outline of the talk allotropes that were discovered in 1985 by researchers at Introduction/History ● History Rice University. ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 3/28

History ■ In chemistry, a fullerene refers to a family of carbon ● Outline of the talk allotropes that were discovered in 1985 by researchers at Introduction/History ● History Rice University. ● General Graph Theory Terms ● More Terms ■ Fullerenes are named after Buckminster Fuller, and are ● . . . More terms ● Independent Sets sometimes called buckyballs (the state molecule of Texas). What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 3/28

History ■ In chemistry, a fullerene refers to a family of carbon ● Outline of the talk allotropes that were discovered in 1985 by researchers at Introduction/History ● History Rice University. ● General Graph Theory Terms ● More Terms ■ Fullerenes are named after Buckminster Fuller, and are ● . . . More terms ● Independent Sets sometimes called buckyballs (the state molecule of Texas). What is a fullerene? ■ The structure of a fullerene is very similar to that of graphite, Counting and Coloring Lemmas which is composed of a sheet of hexagonal rings. Path Lemmas Main Result A tangible result Acknowledgments - p. 3/28

History ■ In chemistry, a fullerene refers to a family of carbon ● Outline of the talk allotropes that were discovered in 1985 by researchers at Introduction/History ● History Rice University. ● General Graph Theory Terms ● More Terms ■ Fullerenes are named after Buckminster Fuller, and are ● . . . More terms ● Independent Sets sometimes called buckyballs (the state molecule of Texas). What is a fullerene? ■ The structure of a fullerene is very similar to that of graphite, Counting and Coloring Lemmas which is composed of a sheet of hexagonal rings. Path Lemmas ■ However, fullerenes contain pentagonal rings that prevent Main Result the sheet from being planar. A tangible result Acknowledgments - p. 3/28

General Graph Theory Terms ■ A graph G is a triple consisting of a vertex set V , an edge ● Outline of the talk set E , and a relation that associates with each edge, two Introduction/History ● History vertices called endpoints . ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 4/28

General Graph Theory Terms ■ A graph G is a triple consisting of a vertex set V , an edge ● Outline of the talk set E , and a relation that associates with each edge, two Introduction/History ● History vertices called endpoints . ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result ■ A simple graph is a graph having no loops or multiple edges. Acknowledgments A loop is an edge whose endpoints are equal. Multiple edges are edges having the same pair of endpoints. - p. 4/28

General Graph Theory Terms ■ A graph G is a triple consisting of a vertex set V , an edge ● Outline of the talk set E , and a relation that associates with each edge, two Introduction/History ● History vertices called endpoints . ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result ■ A simple graph is a graph having no loops or multiple edges. Acknowledgments A loop is an edge whose endpoints are equal. Multiple edges are edges having the same pair of endpoints. ■ Two vertices u and v are said to be adjacent if they are joined by and edge. In this case, u and v are neighbors . - p. 4/28

More Terms ■ If vertex v is the endpoint of an edge e , then we say that v ● Outline of the talk and e are incident Introduction/History ● History ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 5/28

More Terms ■ If vertex v is the endpoint of an edge e , then we say that v ● Outline of the talk and e are incident Introduction/History ● History ■ The valency , or degree of a vertex v is the number of edges ● General Graph Theory Terms ● More Terms the vertex is incident to, denoted deg ( v ) . ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 5/28

More Terms ■ If vertex v is the endpoint of an edge e , then we say that v ● Outline of the talk and e are incident Introduction/History ● History ■ The valency , or degree of a vertex v is the number of edges ● General Graph Theory Terms ● More Terms the vertex is incident to, denoted deg ( v ) . ● . . . More terms ● Independent Sets ■ A planar graph is a graph that can be drawn so that there What is a fullerene? are no edge crossings. Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 5/28

. . . More terms A walk is a consecutive list of incident vertices and edges. A ● Outline of the talk path is a walk with no repeated vertices. Introduction/History ● History ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 6/28

. . . More terms A walk is a consecutive list of incident vertices and edges. A ● Outline of the talk path is a walk with no repeated vertices. Introduction/History ● History ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 6/28

. . . More terms A walk is a consecutive list of incident vertices and edges. A ● Outline of the talk path is a walk with no repeated vertices. Introduction/History ● History ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 6/28

. . . More terms A walk is a consecutive list of incident vertices and edges. A ● Outline of the talk path is a walk with no repeated vertices. Introduction/History ● History ● General Graph Theory Terms ● More Terms ● . . . More terms ● Independent Sets What is a fullerene? Counting and Coloring Lemmas Path Lemmas Main Result A tangible result Acknowledgments - p. 6/28