By Harish Bhattarai, Cameron Halvey, Layton Miller Graph Theory By Harish Bhattarai, Cameron Halvey, Layton Miller

History of Graph Theory Seven Bridges of Königsberg First paper written by Leonhard Euler Published in 1736 Four Color Problem Proposed by Francis Guthrie in 1852 Heinrich Heesch developed methods for proving theorem during 1960-70s Proved by Kenneth Appel and Wolfgang Haken in 1976 Traveling Salesman Problem Origins of the problem are unclear Formulated by William Rowan Hamilton in 1800s Remains unsolved

Connections with MATH 052 A graph is an object consisting of two sets The vertex set is a finite, non-empty set with elements called vertices e.g. 𝑉={1,2,3,…,𝑣} where 𝑣∈ℕ The edge set is a finite set that may be ∅ Its elements are two-element subsets of the vertex set e.g. 𝐸= 1,2 , 1,3 ,…, 𝑣 1 , 𝑣 2 where 𝑣 1 , 𝑣 2 ∈𝑉 ∧ 𝑣 1 ≠ 𝑣 2 Complements of graphs have the same vertex set but opposite edge sets The Seven Bridges of Königsberg can be proved by contradiction

Common Graph Types Cyclic graphs are denoted Cv The vertex set 𝑉= 1,2,3,…,𝑣 where 𝑣∈ℕ such that 𝑣≥3 The edge set 𝐸={{1,2},{2,3},{3,4},…,{𝑣−1,𝑣},{𝑣,1}} Null graphs are denoted by Nv The vertex set 𝑉={1,2,3,…,𝑣} where 𝑣∈ℕ This graph has no edge set Complete graphs are denoted Kv The edge set contains every combination of connections Note that Nv and Kv are complements of each other

Four Color Theorem This problem poses the question: What is the least number of colors needed to color a map such that no two of the same colors touch? It has been proven that four colors are needed to fill any map

Seven Bridges Problem This problem proposes the following question: Given seven bridges scattered over four land masses, can you cross each bridge only once? Proof: Let 𝑉={𝐴,𝐵,𝐶,𝐷} and 𝐸= 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑔 . When crossing a bridge from A to B, we write this as AB. If we then go from B to D, we write this path as ABD, etc. We notice that the number of bridges crossed is one less than the number of letters in the path. So, the number of letters we need in the path is eight. A & B are next to each other twice because there are two bridges connecting A & B. Similarly for A & C, each of A & D, B & D, and C & D must occur together once.

Proof Continued In general, if the number of bridges is any odd number, then the occurrences of A is one half the number of bridges plus one. Therefore there must be three occurrences of A in the path because five bridges lead to A. Next, B must occur twice because there are three bridges leading to B. Similarly, both C and D must occur twice each. However, since this is nine letters and we are only allowed eight as previously shown, it is impossible to cross every bridge only once.

Generalization of the proof Valency (or degree) is the number of edges connected to a vertex. A Eulerian Path is a path that contains each edge of the graph once and only once. Theorem: A graph has an Eulerian Path if and only if the number of vertices of odd valency is either 0 or 2.

Homework Draw the following graphs: C7 N6 K6 Which of these graphs are complements of each other? Given vertex set 𝑉={1,2,3,4,5,6} and edge set 𝐸={ 1,2 , 1,3 , 2,3 , 5,6 }, draw a graph with the vertex set V and edge set E.

References Trudeau, Richard J. Introduction to Graph Theory. New York: Dover Publications, 1993. Print. Wilson, Robin J., and Ian Stewart. Four Colors Suffice How the Map Problem Was Solved. Princeton: Princeton UP, 2014. Print. Wilson, Robin J., William O. James, and E. K. Lloyd. Graph Theory 1736-1936. New York: Oxford UP, 1976. Print.