Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs Maia Wichman Grand Valley State University Partner: Hannah Critchfield, James Madison University Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Overview Background 1 Earth-Moon Problem 2 Specific Cases of Earth-Moon Graphs 3 Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices. A graph is planar if it can be drawn on the plane without edges crossing. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices. A graph is planar if it can be drawn on the plane without edges crossing. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A graph G is a set of vertices V together with a set of edges E which are pairs of vertices. A graph is planar if it can be drawn on the plane without edges crossing. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A proper vertex coloring of a graph is an assignment of colors to the vertices of the graph such that no two adjacent vertices receive the same color. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A proper vertex coloring of a graph is an assignment of colors to the vertices of the graph such that no two adjacent vertices receive the same color. Graph G Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A proper vertex coloring of a graph is an assignment of colors to the vertices of the graph such that no two adjacent vertices receive the same color. Coloring of G Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs A proper vertex coloring of a graph is an assignment of colors to the vertices of the graph such that no two adjacent vertices receive the same color. Coloring of G The chromatic number of a graph G , denoted X ( G ), is the minimum number of colors needed for a vertex coloring of G . Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Theorem (Four Color Theorem, Appel and Haken, 1976) Every planar graph is 4-colorable. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Theorem (Four Color Theorem, Appel and Haken, 1976) Every planar graph is 4-colorable. In other words, the chromatic number of the family of planar graphs is 4. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Theorem (Four Color Theorem, Appel and Haken, 1976) Every planar graph is 4-colorable. In other words, the chromatic number of the family of planar graphs is 4. K 4 : 4 - chromatic Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs While every planar graph is 4-colorable, not every planar graph needs 4 colors. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs While every planar graph is 4-colorable, not every planar graph needs 4 colors. Graph G Best coloring of G uses 3 colors. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs In 1959, Ringel proposed an extension of coloring planar graphs: the Earth-Moon Problem. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs In 1959, Ringel proposed an extension of coloring planar graphs: the Earth-Moon Problem. The motivation for studying the Earth-Moon Problem was map making. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs In 1959, Ringel proposed an extension of coloring planar graphs: the Earth-Moon Problem. The motivation for studying the Earth-Moon Problem was map making. 1 each country on Earth forms one colony on the Moon Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs In 1959, Ringel proposed an extension of coloring planar graphs: the Earth-Moon Problem. The motivation for studying the Earth-Moon Problem was map making. 1 each country on Earth forms one colony on the Moon 2 adjacent areas on the Earth or the Moon receive different colors Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs In 1959, Ringel proposed an extension of coloring planar graphs: the Earth-Moon Problem. The motivation for studying the Earth-Moon Problem was map making. 1 each country on Earth forms one colony on the Moon 2 adjacent areas on the Earth or the Moon receive different colors 3 a country on Earth and its colony on the Moon receive the same color Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs An Earth-Moon Graph is a graph that can be represented as the union of two planar graphs on the same set of vertices. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs An Earth-Moon Graph is a graph that can be represented as the union of two planar graphs on the same set of vertices. G 1 = ( V 1 , E 1 ) G 2 = ( V 2 , E 2 ) The union is G 1 ∪ G 2 := ( V 1 ∪ V 2 , E 1 ∪ E 2 ). Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs How do we color Earth-Moon graphs? What is the chromatic number of the family of Earth-Moon graphs? Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs How do we color Earth-Moon graphs? What is the chromatic number of the family of Earth-Moon graphs? Theorem (Heawood, 1890) Every Earth-Moon graph is 12-colorable. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs
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