PS01-100121-solutions

# PS01-100121-solutions - MATH 682 Problem Set#1 Solutions...

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Unformatted text preview: MATH 682 Problem Set #1 Solutions 1. (10 points) Prove that if graph G is connected and contains a cycle, then there is an edge e in G such that G- e is still connected. Let a cycle in G be denoted by the adjacencies v 1 ∼ v 2 ∼ v 3 ∼ v 4 ∼ ··· ∼ v k ∼ v 1 . Let e be the edge { v 1 ,v 2 } (in principle any edge in the cycle will work; we choose a specific one to simplify the argument). For arbitrary vertices u and v in G , connectivity of G means there is a path u = u 1 ∼ u 2 ∼ u 3 ∼ u 4 ∼ ··· ∼ u ‘ = v . If this path does not traverse the edge e , then u and v are connected in G- e by this path; on the other hand, if the edge e occurs on this path, then we know that there is exactly one i such that { u i ,u i +1 } = e = { v 1 ,v 2 } . There are two possibilities: either u i = v 1 and u i +1 = v 2 , or vice versa. These are handled nearly identically. In the first case, we consider the walk: u = u 1 ∼ u 2 ∼ ··· ∼ u i = v 1 ∼ v k ∼ v k- 1 ∼ ··· ∼ v 2 = u i +1 ∼ u i +2 ∼ ··· ∼ u ‘ = v And in the second: u = u 1 ∼ u 2 ∼ ··· ∼ u i = v 2 ∼ v 3 ∼ ··· ∼ v k ∼ v 1 = u i +1 ∼ u i +2 ∼ ··· ∼ u ‘ = v In both cases we replace the traversal of the edge e in a path from u to v with a traversal of the path around the cycle the long way. Note that this is not necessarily a path; splicing together paths does not guarantee non-self-intersection, but a walk is sufficient to demonstrate connectedness. It is easy to see that e does not appear in this walk: the first and third sections consist of all edges in the path from u to v except for e , and the middle section consists of every edge in the cycle except for e . Thus, u is connected to v in G- e , and since u and v were arbitrary, G- e is connected. 2. (15 points) Recall that α ( G ) , ω ( G ) , δ ( G ) , and Δ( G ) are the independence number, clique number, minimum degree, and maximum degree of G respectively: (a) (5 points) Prove that ω ( G ) ≤ Δ( G ) + 1 and that α ( G ) ≤ | G | - δ ( G ) ....
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## This note was uploaded on 01/12/2012 for the course MATH 682 taught by Professor Wildstrom during the Spring '09 term at University of Louisville.

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PS01-100121-solutions - MATH 682 Problem Set#1 Solutions...

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