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Chapter 5 Euler Circuits Leonhard Euler (1707 - 1783) .. While visiting the town of Konigsberg, in Eastern Europe, Euler was posed with a puzzle.. Can a person walk around town in a way that each of the 7 bridges is crossed only once? So in solving this puzzle, Euler laid the foundations for a new type of geometry, which developed into one of the most practical branches of mathematics -graph theory . Unicursal tracings - tracing a drawing without lifting the pencil or retracing any of the lines. Closed Unicursal tracing - when you end in the same place you started. Open Unicursal tracing - when you start and end in different places. Graph: a picture consisting ofdots, called vertices, and lines, called edges. *The edges do not have to be straight lines, but they must connect two vertices. A graph is a structure that defines pairwise relationships within a set of objects. The objects are vertices, and the pairwise relationships are the edges. so X is related to Y if and only if XY is an edge. Loop - An edge connecting a vertex back with itself. A Example: B E C D In this graph, there are 5 vertices and 6 edges. The vertex set is V = {A, B, C, D, E} The edge set is E = {AB, AD, BB, BD, CD, CD} Notice the edge BB is a loop. Notice the vertex E is an isolated vertex. Notice the graph has more than one edge connecting C and D. The graph is said to have multiple edges. A B E C D The graph is not defined by the picture. The graph is defined by its vertex set and edge set. A B C D E Graph Concepts and Terminology Adjacent Vertices - two vertices that have an edge joining them. A B E C D For Example: Vertices A and D are adjacent. Vertex B is adjacent to itself. Adjacent Edges - two edges that share a common vertex. A B E C For Example: AD and AB are adjacent edges D Degree of a vertex - the number of edges at that vertex. (Loops contributes twice toward the degree.) A B E C D For Example: deg(A) = 2 deg(B) = 4 deg(C) = deg(D) = deg(E) = Odd and Even Vertices - An odd vertex is a vertex of odd degree; an even vertex is a vertex of even degree Path - a sequence of vertices with the property that each vertex in the sequence is adjacent to the next one. It describes a sequence of adjacent edges - a trip along the edges of the graph. *An edge can be a part of a path only once. Length of the Path - the number of edges in the path. More Graph Terminology Circuit - same as a path, but has the requirement that the trip starts and ends at the same vertex. A B E C D For Example: A B D A is a circuit of length 3. This same circuit can be described as B D A B or as D A B D Connected graphs - A graph is connected if, given two vertices, there is a path joining them. A graph that is not connected is called disconnected. A disconnected graph is made up of separate connected components. A B C D This is a disconnected graph E F G H A B C D E G H This is a connected graph F Bridges - An edge that, if erased, the graph would become disconnected. A B E F C D G H For Example: CH is a bridge Euler Path - A path that passes through every edge of a graph. Edges can only be traveled once. Note: a disconnected graph cannot have a Euler path Euler Circuit - A circuit that passes through every edge of a graph. *An Euler Circuit is a closed unicursal tracing of the edges of the graph. * Having a Euler circuit and having an Euler path are mutually exclusive - if a graph has an Euler circuit it cannot have an Euler path, and vice versa.