sol5 - MACM 101 — Discrete Mathematics I Outline...

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MACM 101 — Discrete Mathematics I Outline Solutions to Exercises on Graph Theory 1. Use graph invariants to show that the following graphs are not isomorphic, or find a homomorphism between them. The graphs are isomorphic, and the following labeling of vertices gives an isomorphism. a b c d e f g h a b c d e f g h 2. Show that every connected graph with n vertices has at least n - 1 edges. By induction on the number of vertices. Let P ( n ) denote the statement ‘Every connected graph with n vertices has at least n - 1 edges’. Basis step. Verify P (2) . Every connected graph with 2 vertices contains at least one edge, because otherwise it is a pair of isolated vertices. Hence P (2) is true. Inductive step. Suppose P ( m ) is true for all m k ; we prove P ( k + 1) . Take any connected graph G with k +1 vertices and pick its vertex v . Let G be the graph obtained from G by removing v and all edges incident to it. G can be disconnected, so let G 1 , G 2 , . . . , G s be the connected components of G , and let k i be the number of vertices in G i . By the inductive hypothesis, the number of edges in G i is at least k i - 1 , and the number of edges in G is ( k 1 - 1) + ( k 2 - 1) + . . . + ( k s - 1) = k - s. Observe that there is an edge from v to some vertex in each component G i , because otherwise G i would be disconnected from the rest of the graph G , as well, while G is connected. Therefore the number of edges in G is at least ( k - s ) + s = k = ( k
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sol5 - MACM 101 — Discrete Mathematics I Outline...

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