Fundamentals of Discrete Math for Computer Science by Tom Jenkyns & Ben Stephenson

Author:Tom Jenkyns & Ben Stephenson
Language: eng
Format: epub
Publisher: Springer London, London

// X If π = (v 0, e 1, v 1, …, e k , v k ) is a closed path and k is odd,

//then the graph G π contains a polygon.

Lemma 5.2.3:

If there are two distinct simple paths in G joining the same two vertices, then G contains a polygon.

Proof

Suppose

are two distinct simple paths in G joining the same two vertices, y and z.

Compare the sequences π1 and π2 entry by entry. They both begin with the same vertex, y. If f 1 = e 1, then w 1 must equal v 1; then if f 2 = e 2, then w 2 must equal v 2. But π1 and π2 are different sequences so there must be an entry where they differ, and the first entry where they differ must be an edge. Thus, there is an index j such that 0 <= j < k where (as sequences) // Why is j < k ?

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