# 4204hw1 - G A set of vertices S in the graph G is called a vertex cover if all edges of G have at least one of their vertices in S Let τ G denote

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Combinatorics II (MAD 4204) — Spring 2011 Due Friday 1/21/2011 Homework #1 Please note that all answers require justifcation/prooF. Also, my o±ce hours are: M5 (11:45–12:35), T8 (3:00–3:50), W5 (11:45–12:35), and by appointment in general, although we don’t have class on Monday, 1/17/2011. 1 Let G be the graph obtained from the complete graph K n by deleting an edge. Find a formula for the number of spanning trees of G . 2 Let G be the graph obtained from the complete bipartite graph K m,n by deleting an edge. Find a formula for the number of spanning trees of
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Unformatted text preview: G . A set of vertices S in the graph G is called a vertex cover if all edges of G have at least one of their vertices in S . Let τ ( G ) denote the size of the smallest vertex cover of G and let ν ( G ) denote the size of its maximum matching. 3 Prove that in any graph G , the inequality ν ( G ) ≤ τ ( G ) holds. 4 Prove that in any bipartite graph G , we actually have ν ( G ) = τ ( G )....
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## This note was uploaded on 12/10/2011 for the course MAD 4204 taught by Professor Vetter during the Spring '11 term at University of Florida.

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