![Page 1: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/1.jpg)
Graph TheoryKatherine and Noam
![Page 2: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/2.jpg)
The Seven Bridges of Königsberg
![Page 3: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/3.jpg)
The Seven Bridges of Königsberg
● Königsberg is a city in Russia● It is separated by several rivers, and there are bridges
over these rivers
Image credit: Bogdan Giuşcă - Public domain (PD), based on the image, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=112920
![Page 4: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/4.jpg)
The Seven Bridges of Königsberg
● Is there a way to cross each of these bridges exactly once?
![Page 5: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/5.jpg)
The Seven Bridges of Königsberg: Activity!
● Is there a way to cross each of these bridges exactly once?
● Try it!
![Page 6: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/6.jpg)
The Seven Bridges of Königsberg
● Is there a way to cross each of these bridges exactly once? NO!
![Page 7: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/7.jpg)
The Seven Bridges of Königsberg
● Is there a way to cross each of these bridges exactly once? NO!
● How can we understand this mathematically?
![Page 8: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/8.jpg)
Rephrase the problem!
● Let’s think of each land mass as a point● Think of the bridges as edges between the points
Image credit: https://physics.weber.edu/carroll/honors/konigsberg.htm
![Page 9: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/9.jpg)
Graph theory!
● This kind of representation of our problem is a graph.
Image credit: Wikipedia
![Page 10: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/10.jpg)
Wait, what’s a graph?
![Page 11: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/11.jpg)
What is a graph?
● Sometimes when people say “graph”, they mean something like this or this:
Image credit: https://www.podfeet.com/blog/tutorials-5/how-to-make-2d-excel-graphs-look-3d/, http://csas.ei.columbia.edu/2016/09/26/a-better-graph/
![Page 12: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/12.jpg)
What is a graph?
● But often, mathematicians mean something like this or this:
Image credit: https://www.maths.unsw.edu.au/about/graph-theory-unsw-past-and-present, Wikipedia
![Page 13: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/13.jpg)
What is a graph?
● More formally, a graph is just a set of points (vertices) and edges connecting those points.
● The points can be labeled (but don’t have to be)
![Page 14: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/14.jpg)
Graphs in the real world
![Page 15: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/15.jpg)
Eulerian Paths
![Page 16: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/16.jpg)
Eulerian Path
● An Eulerian path is a path through a graph that travels through each edge exactly once.
![Page 17: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/17.jpg)
Eulerian Path
● An Eulerian path is a path through a graph that travels through each edge exactly once.
● The Bridges of Königsberg is really asking whether we can find an Eulerian path through this graph.
![Page 18: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/18.jpg)
Eulerian Path: Activity!
● An Eulerian path is a path through a graph that travels through each edge exactly once.
● Let’s understand when we can and can’t find an Eulerian path through a graph.
![Page 19: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/19.jpg)
Eulerian Path Recap
● If we have an Eulerian path, what vertices can have an odd number of edges coming from them?
○ The start vertex○ The end vertex
![Page 20: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/20.jpg)
Planar Graphs
![Page 21: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/21.jpg)
Graph theory
● Question: What information determines a graph?
![Page 22: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/22.jpg)
Graph theory
● Question: What information determines a graph?● Answer: Just the vertices (points) and the edges (connections
between points). Nothing else matters!
![Page 23: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/23.jpg)
Graph theory
● Question: What information determines a graph?● Answer: Just the vertices (points) and the edges (connections
between points). Nothing else matters!
Are these the same graph?
![Page 24: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/24.jpg)
Graph theory
● Question: What information determines a graph?● Answer: Just the vertices (points) and the edges (connections
between points). Nothing else matters!
What about these?
![Page 25: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/25.jpg)
Graph theory
● These look really different, but are all the same graph
Image credit: http://mathworld.wolfram.com/PetersenGraph.html
![Page 26: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/26.jpg)
Planar graphs
● If we can draw a graph without its edges crossing, then the graph is planar
![Page 27: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/27.jpg)
Planar graphs
● If we can draw a graph without its edges crossing, then the graph is planar
● Example:
Image credit: http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/planarity.htm
![Page 28: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/28.jpg)
Planar graphs
● If we can draw a graph without its edges crossing, then the graph is planar
● Example:
The edges are crossing right now, but can we redraw it?
Image credit: http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/planarity.htm
![Page 29: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/29.jpg)
Planar graphs
● If we can draw a graph without its edges crossing, then the graph is planar
● Example:
Image credit: http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/planarity.htm
![Page 30: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/30.jpg)
Planar graphs
● If we can draw a graph without its edges crossing, then the graph is planar
● Now you try!
Image credit: http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/planarity.htm
![Page 31: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/31.jpg)
Graph Theory and Coloring
![Page 32: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/32.jpg)
Map Coloring
Fill in every region so that no two adjacent regions have the same color.
![Page 33: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/33.jpg)
How many colors would it take to color this map?
![Page 34: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/34.jpg)
You can do it with four!
Image credit: Wikipedia
![Page 35: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/35.jpg)
Now, challenge each other...
Pair up, and make a challenge map for your partner that you think should require many colors.
Then switch maps and try to color your partner’s challenge map with as few colors as possible!
![Page 36: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/36.jpg)
The Four Color Theorem
Any map can be colored with just four colors.
Very hard to prove!
![Page 37: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/37.jpg)
The Five Color Theorem Any map can be colored
with five colors.
![Page 38: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/38.jpg)
From maps to graphs
● Do you think we could represent this map as a graph?
![Page 39: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/39.jpg)
From maps to graphs
● Regions -> points, edges connect adjacent regions
Like this!
![Page 40: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/40.jpg)
From maps to graphs
If we represent a map as a graph, is it planar?
![Page 41: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/41.jpg)
From maps to graphs
● If we represent a map as a graph, is it planar?● YES!
![Page 42: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/42.jpg)
The Five Color Theorem
We can restate the result:
Every planar graph can be colored with five colors so that any two vertices connected by an edge have different colors.
Using graph theory language!
![Page 43: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/43.jpg)
Why is this true?
● What about for planar graphs with at most 5 vertices?
![Page 44: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/44.jpg)
Why is this true?
● What about for planar graphs with at most 5 vertices?● Just make each vertex a different color!
![Page 45: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/45.jpg)
Why is this true?
● Now, let’s consider more general planar graphs.● A fact from math: Every planar graph has a vertex that’s
connected to at most 5 edges.
![Page 46: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/46.jpg)
Why is this true?
● Now, let’s consider more general planar graphs.● A fact from math: Every planar graph has a vertex that’s
connected to at most 5 edges.● What if we’ve colored our graph except for that vertex?
![Page 47: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/47.jpg)
Why is this true?
● In other words, we’ve colored our graph except for one vertex that’s connected to at most 5 edges.
![Page 48: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/48.jpg)
Why is this true?
● We’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to 4 edges?
Image credit: https://astarmathsandphysics.com/university-maths-notes/topology/2389-the-four-colour-mapping-theorem.html
![Page 49: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/49.jpg)
Why is this true?
● We’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to exactly 5 edges?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
![Page 50: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/50.jpg)
Why is this true?
● We’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to exactly 5 edges?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
Can we make the red vertex yellow?
![Page 51: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/51.jpg)
Why is this true?
● Let’s say we’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to exactly 5 edges?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
What if they’re connected?
![Page 52: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/52.jpg)
Why is this true?
● Let’s say we’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to exactly 5 edges?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
Can we make the green vertex yellow?
![Page 53: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/53.jpg)
Why is this true?
● Let’s say we’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to exactly 5 edges?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
What if they’re connected?
![Page 54: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/54.jpg)
Why is this true?
● Let’s say we’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to exactly 5 edges?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
Can we make the blue vertex red?
![Page 55: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/55.jpg)
Why is this true?
● Let’s say we’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to exactly 5 edges?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
What if they’re connected?
![Page 56: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/56.jpg)
Why is this true?
● Let’s say we’ve colored our graph except for one vertex that’s connected to at most 5 edges.
○ What if it’s only connected to exactly 5 edges?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
What if they’re connected?
THEY CAN’T BE--our graph is planar.
![Page 57: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/57.jpg)
Why is this true?
● Can we always either make the green vertex yellow OR make the red vertex blue?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
We don’t know what the rest of our graph looks like!
![Page 58: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/58.jpg)
Why is this true?
● The only time we can’t make the green vertex yellow is when something like this happens...
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
![Page 59: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/59.jpg)
Why is this true?
● The only time we can’t make the green vertex yellow is when something like this happens...
● And then, since our graph is planar, we will be able to make the red vertex blue!
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
![Page 60: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/60.jpg)
Why is this true?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
● We’ve colored our graph except for one vertex that’s connected to exactly 5 edges.
● We can color the neighboring vertices with just 4 colors...
![Page 61: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/61.jpg)
Why is this true?
Image credit: http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/MatthewWahab/5color.html
● We’ve colored our graph except for one vertex that’s connected to exactly 5 edges.
● We can color the neighboring vertices with just 4 colors... and the middle vertex with the fifth.
![Page 62: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/62.jpg)
Mathematical induction
● We know that a graph with 1 vertex can be 5-colored ● We know that if we can 5-color a graph with n vertices,
then we can 5-color a graph with n+1 vertices (why?)
![Page 63: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/63.jpg)
Mathematical induction
● We know that a graph with 1 vertex can be 5-colored ● We know that if we can 5-color a graph with n vertices,
then we can 5-color a graph with n+1 vertices (why?)● By induction, this proves the 5-color theorem!
![Page 64: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/64.jpg)
Context and Applications
![Page 65: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/65.jpg)
You can do lots of other things with graphs...
● We’ve focused on undirected, unweighted graphs● But you could have graphs with directed edges
Image credit: Wikimedia
![Page 66: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/66.jpg)
You can do lots of other things with graphs...
● We’ve focused on undirected, unweighted graphs● But you could have graphs with weighted edges
Image credit: http://pages.cpsc.ucalgary.ca/~jacobs/Courses/cpsc331/F08/tutorials/tutorial14.html
![Page 67: Graph Theory - Carnegie Mellon School of Computer Sciencekcordwel/GraphTheoryandColoring.pdfGraph theory Question: What information determines a graph? Answer: Just the vertices (points)](https://reader033.vdocument.in/reader033/viewer/2022051807/60063fa06bb8ab0c1724bc86/html5/thumbnails/67.jpg)
Graph Theory Uses
● Modeling○ Social networks○ Rumor spreading○ Disease transmission○ Molecules○ Atomic structures
● Advanced math○ Knot theory○ Group theory