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(a) Show that a connected graph with at least as many edges as vertices must contain a cycle.

(a) Show that a connected graph with at least as many edges as vertices must contain a cycle.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter10: Analytic Geometry
Section10.4: Analytic Proofs
Problem 29E: Based on the result in Exercise 26, describe the graph of the equation x2+y2=16.
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(a) Show that a connected graph with at least as many edges as vertices must contain a cycle. (b) Let G be a connected graph with n vertices and m edges for n ≥ 2. Show that G contains a bipartite subgraph H = H(V, E) with the same vertex set as G (i.e V (H) = V(G)) and |E(H)| ≥ 2 m you should be able to give a construction of this graph, m 2 proving why it has at least edges. (c) Let K be a connected graph with n vertices and at least 2n - 1 edges for n>2. Show that K must contain a cycle of even length.
Transcribed Image Text:(a) Show that a connected graph with at least as many edges as vertices must contain a cycle. (b) Let G be a connected graph with n vertices and m edges for n ≥ 2. Show that G contains a bipartite subgraph H = H(V, E) with the same vertex set as G (i.e V (H) = V(G)) and |E(H)| ≥ 2 m you should be able to give a construction of this graph, m 2 proving why it has at least edges. (c) Let K be a connected graph with n vertices and at least 2n - 1 edges for n>2. Show that K must contain a cycle of even length.
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