Eulerian Cycle (2A) Young Won Lim 4/26/18

Transcription

1 Eulerian Cycle (2A)

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using LibreOffice and Octave.

3 Euler Cycle visits every edge exactly once a necessary condition for the existence of Eulerian cycles is that all vertices in the graph have an even degree that connected graphs with all vertices of even degree have an Eulerian circuit. Graph (5A) 3

4 Euler Cycle All odd degree vertices All even degree vertices Graph (5A) 4

5 Euler Cycle F B H E D C I J G ABCDEFGHIJK A K en.wikipedia.org Graph (5A) 5

6 Eulerian Cycles of Undirected Graphs An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single connected component An undirected graph can be decomposed into edge-disjoint cycles if and only if all of its vertices have even degree. So, a graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint cycles and its nonzero-degree vertices belong to a single connected component Graph (5A) 6

7 Edge Disjoint Cycle Decomposition F B H E C I G D J A K Graph (5A) 7

8 Eulerian Paths of Undirected Graphs An undirected graph has an Eulerian path if and only if exactly zero or two vertices have odd degree, and all of its vertices with nonzero degree belong to a single connected component. Graph (5A) 8

9 Eulerian Cycles of Directed Graphs A directed graph has an Eulerian cycle if and only if every vertex has equal in degree and out degree, and all of its vertices with nonzero degree belong to a single strongly connected component. Equivalently, a directed graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint directed cycles and all of its vertices with nonzero degree belong to a single strongly connected component. Graph (5A) 9

10 Eulerian Paths of Directed Graphs A directed graph has an Eulerian path if and only if at most one vertex has (out-degree) (in-degree) = 1, at most one vertex has (in-degree) (out-degree) = 1, every other vertex has equal in-degree and out-degree, and all of its vertices with nonzero degree belong to a single connected component of the underlying undirected graph. Graph (5A) 10

11 Seven Bridges of Königsberg The problem was to devise a walk through the city that would cross each of those bridges once and only once. Graph (5A) 11

12 Seven Bridges of Königsberg 3 3 A B C A B C 5 D 3 5 D 4 E F G E F G H 3 4 Eulerian Path AEHGFDCB Graph (5A) 12

13 Seven Bridges of Königsberg 4 4 A B C A B C 5 I D 4 6 I D 4 E F G E J F G H H 5 6 Eulerian Path EHGFDCBAI Eulerian Cycle AEHGFDCBAJI Graph (5A) 13

14 Euler Cycle Any connected graph with even degree vertices An Euler cycle A proof by induction on the number of edges in G A connected graph G with even degree vertices only and k edges (k < n) An Euler cycle Assume this is true A connected graph G with even degree vertices only and n edges An Euler cycle Then this holds true Johnsonbough, Discrete Mathematics Graph (5A) 14

15 Euler Cycle Any connected graph with even degree vertices which has n edge An Euler cycle Any connected graph with even degree vertices which has n-1 edge Any connected graph with even degree vertices which has n-2 edge An Euler cycle An Euler cycle Any connected graph with even degree vertices which has 2 edge Any connected graph with even degree vertices which has 1 edge An Euler cycle An Euler cycle Johnsonbough, Discrete Mathematics Graph (5A) 15

16 Euler Cycle Base Cases n = 0 edge n = 1 edge all even degree vertices n = 2 edge an Euler cycle Johnsonbough, Discrete Mathematics Graph (5A) 16

17 Euler Cycle decrease the number of edges by one A connected graph G with even degree vertices only and n edges (k < n) e 1 e 2 v 1 v 2 v 3 all even degree vertices P: a path from v to v 1 A connected graph G' with even degree vertices only and n-1 edges (k < n) P': a portion of the path P that are in G' e v 1 v 2 v 3 all even degree vertices Johnsonbough, Discrete Mathematics Graph (5A) 17

18 Euler Cycle a path from v to v 1 Case 1: P ends at v 1 Case 2: P ends at v 2 Case 3: P ends at v 3 v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 P P P v v v v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 P ' P ' P ' v v v Johnsonbough, Discrete Mathematics Graph (5A) 18

19 Euler Cycle F B H E D C I J G ABCDEFGHIJK A K en.wikipedia.org Graph (5A) 19

20 Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 2 components Graph (5A) 20

21 Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 1 component Graph (5A) 21

22 Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 2 components Graph (5A) 22

23 Euler Cycle G G' v 1 v 3 v 1 v 3 v 2 v 2 1 component Graph (5A) 23

24 References [1] [2]