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
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 A graph is planar if it can be drawn on the plane without edges crossing.
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 A graph is planar if it can be drawn on the plane without edges crossing.
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 Graph G
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 The chromatic number of a graph G, denoted X(G), is the minimum number of colors needed for a vertex coloring of G.
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 In other words, the chromatic number of the family of planar graphs is 4.
K4: 4 - chromatic
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 K4: 4 - chromatic
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.
K4: 4 - chromatic
Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs Graph G
Best coloring of G uses 3 colors.
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 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
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 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
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 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
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 3 a country on Earth and its colony on the Moon receive the same color
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 G1 = (V1, E1) G2 = (V2, E2) The union is G1 ∪ G2 := (V1 ∪ V2, E1 ∪ E2).
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.
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 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 Theorem (Heawood, 1890) Every Earth-Moon graph is 12-colorable.
The lower bound for X(Earth-Moon graphs) is 9.
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 The lower bound for X(Earth-Moon graphs) is 9.
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 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 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 X(Earth-Moon graphs) = 9, 10, 11, or 12?
Simplify 2 planar layers into 2 cyclic layers.
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 Simplify 2 planar layers into 2 cyclic layers.
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 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 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.
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 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.
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 Theorem (C. and W.)
All doubly cyclic graphs are 4-colorable, except K5.
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 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 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.
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs
A cyclic - outerplanar graph is the union of a cycle and an outerplanar graph on the same set of vertices.
Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs 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.
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs
A cyclic - outerplanar graph is the union of a cycle and an outerplanar 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 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.
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs
A cyclic - outerplanar graph is the union of a cycle and an outerplanar 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 Theorem (C. and W.) Let G be a cyclic - outerplanar graph. Then G is 5-colorable.
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs
A cyclic - outerplanar graph is the union of a cycle and an outerplanar 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 Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs
A cyclic - outerplanar graph is the union of a cycle and an outerplanar 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 • 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.
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 • Key fact: cyclic - outerplanar graphs have at least one vertex with degree at most 4.
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 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 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 Theorem (Gethner and Sulanke) The list of possibilities for the chromatic number of the family of doubly outerplanar graphs is {6,7,8}.
Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs
A doubly outerplanar graph is the union of two outerplanar 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
A doubly outerplanar graph is the union of two outerplanar graphs on 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 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 Theorem (C. and W.) All doubly outerplanar graphs on n vertices where n ≤ 22 are 7-colorable.
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 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 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 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 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 Background Earth-Moon Problem Specific Cases of Earth-Moon Graphs
1 7 6 3
3 1
0 9 2 9
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2 4 0 10
10 5 8 4
Critchfield, Wichman Grand Valley State University REU Exploring The Earth-Moon Problem for Doubly Outerplanar Graphs