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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,

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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

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Overview

1

Background

2

Earth-Moon Problem

3

Specific Cases of Earth-Moon Graphs

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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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

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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

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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

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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

SLIDE 7

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

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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

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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

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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

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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

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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

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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

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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.

K4: 4 - chromatic

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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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

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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

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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

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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

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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

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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

SLIDE 21

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

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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

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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. G1 = (V1, E1) G2 = (V2, E2) The union is G1 ∪ G2 := (V1 ∪ V2, E1 ∪ E2).

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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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

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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

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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

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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

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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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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. The lower bound for X(Earth-Moon graphs) is 9.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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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. The lower bound for X(Earth-Moon graphs) is 9.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Determining X(Earth-Moon graphs) is an open problem in Graph Theory.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Determining X(Earth-Moon graphs) is an open problem in Graph

Theory. X(Earth-Moon graphs) = 9, 10, 11, or 12?

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Determining X(Earth-Moon graphs) is an open problem in Graph

Theory. X(Earth-Moon graphs) = 9, 10, 11, or 12?

Simplify 2 planar layers into 2 cyclic layers.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A doubly cyclic graph is the union of two cycles on the same set of vertices.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A doubly cyclic graph is the union of two cycles on the same set of vertices.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A doubly cyclic graph is the union of two cycles on the same set of vertices. What is the chromatic number of the family of doubly cyclic graphs?

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A doubly cyclic graph is the union of two cycles on the same set of vertices. What is the chromatic number of the family of doubly cyclic graphs? Theorem (C. and W.) All doubly cyclic graphs are 4-colorable, except K5.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A cyclic - outerplanar graph is the union of a cycle and an

uterplanar graph on the same set of vertices.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A cyclic - outerplanar graph is the union of a cycle and an

uterplanar graph on the same set of vertices.

A outerplanar graph is a planar graph in which all vertices belong to the outer face.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A cyclic - outerplanar graph is the union of a cycle and an

uterplanar graph on the same set of vertices.

A outerplanar graph is a planar graph in which all vertices belong to the outer face.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A cyclic - outerplanar graph is the union of a cycle and an

uterplanar graph on the same set of vertices.

A outerplanar graph is a planar graph in which all vertices belong to the outer face. What is the chromatic number of the family of cyclic - outerplanar graphs?

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A cyclic - outerplanar graph is the union of a cycle and an

uterplanar graph on the same set of vertices.

A outerplanar graph is a planar graph in which all vertices belong to the outer face. What is the chromatic number of the family of cyclic - outerplanar graphs? Theorem (C. and W.) Let G be a cyclic - outerplanar graph. Then G is 5-colorable.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Theorem (C. and W.) Let G be a cyclic - outerplanar graph. Then G is 5-colorable.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Theorem (C. and W.) Let G be a cyclic - outerplanar graph. Then G is 5-colorable.

Proof by induction on the number of vertices of G.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Theorem (C. and W.) Let G be a cyclic - outerplanar graph. Then G is 5-colorable.

Proof by induction on the number of vertices of G.
Key fact: cyclic - outerplanar graphs have at least one vertex

with degree at most 4.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Theorem (C. and W.) Let G be a cyclic - outerplanar graph. Then G is 5-colorable.

Proof by induction on the number of vertices of G.
Key fact: cyclic - outerplanar graphs have at least one vertex

with degree at most 4.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A doubly outerplanar graph is the union of two outerplanar graphs

n the same set of vertices.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

A doubly outerplanar graph is the union of two outerplanar graphs

n the same set of vertices.

Theorem (Gethner and Sulanke) The list of possibilities for the chromatic number of the family of doubly outerplanar graphs is {6,7,8}.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Theorem (C. and W.) All doubly outerplanar graphs on n vertices with a layer that has a vertex of degree n − 1 are 6-colorable.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Theorem (C. and W.) All doubly outerplanar graphs on n vertices with a layer that has vertices v and u with deg(v) + deg(u) = n + 2 are 7-colorable.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Theorem (C. and W.) All doubly outerplanar graphs on n vertices with a layer that has vertices v and u with deg(v) + deg(u) = n + 2 are 7-colorable. Theorem (C. and W.) All doubly outerplanar graphs on n vertices where n ≤ 22 are 7-colorable.

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Types of Graphs and Their Chromatic Numbers Graph Type Layers & X(G) Earth - Moon 2 planar X(G) = 9, 10, 11, or 12 Singly Outerplanar 1 planar, 1 outerplanar X(G) = 8, 9, or 10 Doubly Outerplanar 2 outerplanar X(G) = 6, 7, or 8 Cyclic - Outerplanar 1 cyclic, 1 outerplanar X(G) = 5 Doubly Cyclic 2 cyclic X(G) = 5

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

References

Ellen Gethner, Thom Sulanke, Thickness-Two Graphs Part Two: More New Nine-Critical Graphs, Independence Ratio, Cloned Planar Graphs, and Singly and Doubly Outerplanar Graphs, Graphs and Combinatorics 25 (2009) 197-217. Joan P. Hutchinson, Coloring Ordinary Maps, Maps of Empires, and Maps of the Moon, Macalester College (1993) John Harris, Jeffry Hirst, Michael Mossinghoff, Combinatorics and Graph Theory 2nd edition, Springer 233 Spring Street, New York, NY 10013, USA (2011). Kyle F. Jao, Doublas B. West, Vertex Degrees in Outerplanar Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 82 (2012) 229-239. McKay, B.D. and Piperno, A., Practical Graph Isomorphism, II, Journal of Symbolic Computation, 60 (2014), pp. 94-112, http://dx.doi.org/10.1016/j.jsc.2013.09.003 SageMath, the Sage Mathematics Software System (Version 8.2), The Sage Developers, 2018, http://www.sagemath.org. Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs

Acknowledgements

This work was partially supported by National Science Foundation grant DMS-1659113, which funds a Research Experiences for Undergraduates program at Grand Valley State University. Grand Valley State University My partner Hannah Critchfield and Dr. Emily Marshall, our mentor

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs

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Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs