On The Coloring of Graphs and Chromatic Polynomials Ian Cavey, - - PowerPoint PPT Presentation

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On The Coloring of Graphs and Chromatic Polynomials Ian Cavey, Christian Sprague, Mack Stannard Boise State University November 9, 2014 Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 1 /

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On The Coloring of Graphs and Chromatic Polynomials

Ian Cavey, Christian Sprague, Mack Stannard

Boise State University

November 9, 2014

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 1 / 20

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SLIDE 2

Introduction

Introduction

G

  • G is an example of a Graph

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 2 / 20

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Introduction

Introduction

G

  • G is an example of a Graph
  • Graphs are made up of:
  • Vertices

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 3 / 20

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SLIDE 4

Introduction

Introduction

G

  • G is an example of a Graph
  • Graphs are made up of:
  • Vertices
  • Edges

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 4 / 20

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Introduction

Introduction

G

  • Assigning a color to every vertex in a graph is called graph coloring

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 5 / 20

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SLIDE 6

Introduction

Introduction

G

  • Assigning a color to every vertex in a graph is called graph coloring
  • A proper coloring of a graph is a coloring where any two vertices

connected by an edge are not colored identically

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 5 / 20

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Introduction

Introduction

G

  • Assigning a color to every vertex in a graph is called graph coloring
  • A proper coloring of a graph is a coloring where any two vertices

connected by an edge are not colored identically

  • Is the above coloring proper or improper?

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 5 / 20

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Introduction

Introduction

G

  • Assigning a color to every vertex in a graph is called graph coloring
  • A proper coloring of a graph is a coloring where any two vertices

connected by an edge are not colored identically

  • Is the above coloring proper or improper?
  • Improper!

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 6 / 20

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Introduction

Introduction

Natural Questions

  • What is the fewest number of colors needed to properly color a graph?

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 7 / 20

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SLIDE 10

Introduction

Introduction

Natural Questions

  • What is the fewest number of colors needed to properly color a graph?
  • How many ways can a graph be properly colored with x colors?

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 7 / 20

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Chromatic Polynomials and Numbers

Chromatic Polynomials and Numbers

The Chromatic Number is the fewest number of colors needed to properly color a graph.

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 8 / 20

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Chromatic Polynomials and Numbers

Chromatic Polynomials and Numbers

The Chromatic Number is the fewest number of colors needed to properly color a graph.

  • How many ways are there to properly color a graph G with x colors?

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 8 / 20

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Chromatic Polynomials and Numbers

Chromatic Polynomials and Numbers

The Chromatic Number is the fewest number of colors needed to properly color a graph.

  • How many ways are there to properly color a graph G with x colors?
  • We can create a function of a graph G and a number of colors x

which is the number of ways to color G properly with x colors

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 8 / 20

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Chromatic Polynomials and Numbers

Chromatic Polynomials and Numbers

The Chromatic Number is the fewest number of colors needed to properly color a graph.

  • How many ways are there to properly color a graph G with x colors?
  • We can create a function of a graph G and a number of colors x

which is the number of ways to color G properly with x colors

  • This function is actually a polynomial, called the Chromatic

Polynomial, and is denoted F(G, x)

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 8 / 20

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Reduction Algorithm

Reduction Algorithm

  • Finding the Chromatic Polynomial for a graph is not always easy!

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 9 / 20

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Reduction Algorithm

Reduction Algorithm

  • Finding the Chromatic Polynomial for a graph is not always easy!
  • But thanks to Birkhoff and Lewis, we have a simple algorithm for

computing F(G, x) given a graph G.

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 9 / 20

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Reduction Algorithm

Reduction Algorithm

G a b e

  • Consider two vertices a and b that are connected by edge e.

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 10 / 20

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Reduction Algorithm

Reduction Algorithm

G a b e

  • Consider two vertices a and b that are connected by edge e.
  • Since we want proper colorings, a and b must be colored differently

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 10 / 20

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Reduction Algorithm

Reduction Algorithm

G a b e

  • Consider two vertices a and b that are connected by edge e.
  • Since we want proper colorings, a and b must be colored differently
  • To determine the number of ways in which vertices a, b can be

colored differently:

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 10 / 20

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Reduction Algorithm

Reduction Algorithm

G a b e

  • Consider two vertices a and b that are connected by edge e.
  • Since we want proper colorings, a and b must be colored differently
  • To determine the number of ways in which vertices a, b can be

colored differently:

  • Find the number of colorings where a and b are colored the same or

different

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 10 / 20

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Reduction Algorithm

Reduction Algorithm

G a b e

  • Consider two vertices a and b that are connected by edge e.
  • Since we want proper colorings, a and b must be colored differently
  • To determine the number of ways in which vertices a, b can be

colored differently:

  • Find the number of colorings where a and b are colored the same or

different

  • Then subtract the number of ways where a and b are colored

identically.

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 10 / 20

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Reduction Algorithm

Reduction Algorithm

First find the number of colorings where a and b are colored the same or different. G a b e

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 11 / 20

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Reduction Algorithm

Reduction Algorithm

First find the number of colorings where a and b are colored the same or different. G ′ a b

  • Remove edge e

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 12 / 20

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Reduction Algorithm

Reduction Algorithm

First find the number of colorings where a and b are colored the same or different. G ′ a b

  • Remove edge e
  • Call this new graph G ′

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 12 / 20

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Reduction Algorithm

Reduction Algorithm

First find the number of colorings where a and b are colored the same or different. G ′ a b

  • Remove edge e
  • Call this new graph G ′
  • F(G ′, x) is the number of (otherwise proper) colorings of G where a

and b can have the same or different colors.

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 12 / 20

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Reduction Algorithm

Reduction Algorithm

Then subtract the number of ways where a and b are colored identically. G a b e

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 13 / 20

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Reduction Algorithm

Reduction Algorithm

Then subtract the number of ways where a and b are colored identically. G ′′ a,b

  • Merge a and b into one vertex

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 14 / 20

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Reduction Algorithm

Reduction Algorithm

Then subtract the number of ways where a and b are colored identically. G ′′ a,b

  • Merge a and b into one vertex
  • Call this graph G ′′

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 14 / 20

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Reduction Algorithm

Reduction Algorithm

Then subtract the number of ways where a and b are colored identically. G ′′ a,b

  • Merge a and b into one vertex
  • Call this graph G ′′
  • F(G ′′, x) is the number of (otherwise proper) colorings of G where a

and b are colored identically

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 14 / 20

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Reduction Algorithm

Reduction Algorithm

According to the algorithm, the chromatic polynomial of G is equal to the difference of the chromatic polynomials of G ′ and G ′′, F(G, x) = F(G ′, x) − F(G ′′, x).

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 15 / 20

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Reduction Algorithm

Reduction Algorithm

According to the algorithm, the chromatic polynomial of G is equal to the difference of the chromatic polynomials of G ′ and G ′′, F(G, x) = F(G ′, x) − F(G ′′, x).

  • This describes one iteration of the algorithm

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 15 / 20

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Reduction Algorithm

Reduction Algorithm

According to the algorithm, the chromatic polynomial of G is equal to the difference of the chromatic polynomials of G ′ and G ′′, F(G, x) = F(G ′, x) − F(G ′′, x).

  • This describes one iteration of the algorithm
  • Continue in this manner by selecting edges until all of the obtained

graphs are without edges

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 15 / 20

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Reduction Algorithm

Reduction Algorithm

According to the algorithm, the chromatic polynomial of G is equal to the difference of the chromatic polynomials of G ′ and G ′′, F(G, x) = F(G ′, x) − F(G ′′, x).

  • This describes one iteration of the algorithm
  • Continue in this manner by selecting edges until all of the obtained

graphs are without edges

  • For each of these empty graphs with n vertices and x colors there are

xn different colorings, since each of the n vertices has x possible colorings

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 15 / 20

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Results

Results

  • Finally, we introduce some well known types of graphs

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 16 / 20

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Results

Results

  • Finally, we introduce some well known types of graphs
  • Applying the Birkhoff-Lewis Reduction Algorithm and induction to

these graphs yields general formulas for Chromatic Polynomials

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 16 / 20

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Results

Results

Path

A path graph of n vertices, denoted Pn, is a graph that can be arranged in such a way that vertices are only connected to their adjacent vertices P

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 17 / 20

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Results

Results

Path

A path graph of n vertices, denoted Pn, is a graph that can be arranged in such a way that vertices are only connected to their adjacent vertices P

  • The chromatic polynomial for a path graph has a general form,

F(Pn, x) = x(x − 1)n−1

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 17 / 20

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Results

Results

Cycle

A cycle graph of n vertices, denoted Cn, is composed of an equal number

  • f vertices and edges such that each vertex has exactly two edges

connected with it, creating a closed graph.

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 18 / 20

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Results

Results

Cycle

A cycle graph of n vertices, denoted Cn, is composed of an equal number

  • f vertices and edges such that each vertex has exactly two edges

connected with it, creating a closed graph.

  • The chromatic polynomial for a cycle graph has a general form,

F(C, n) = (x − 1)n + (−1)n+1(1 − x)

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 18 / 20

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Results

Results

Complete

A graph with n vertices where any two vertices are connected by an edge is called a complete graph, denoted Kn.

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 19 / 20

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Results

Results

Complete

A graph with n vertices where any two vertices are connected by an edge is called a complete graph, denoted Kn.

  • The chromatic polynomial for a complete graph has a general form,

F(Kn, x) = x(x − 1)(x − 2) · · · (x − n + 1) =

x! (x−n)!.

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 19 / 20

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Results

Acknowledgments Thanks to Dr. Andr´ es Caicedo, Boise State University, for providing the Sage code, which enabled us to explore and make conjectures Bibliography Mount Holyoke College Laboratories in Mathematical Experimentation: A Bridge to Higher Mathematics 1997: Springer-Verlag New York, Inc. www.sagemathcloud.com

Ian Cavey, Christian Sprague, Mack Stannard (Boise State University) Coloring Slides November 9, 2014 20 / 20