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# ass9-new - e then each component of G-e is bipartite and so...

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Due Friday July 15 Math 239 Assignment 9 1. A subset S of the vertices of a graph is independent if no two vertices in it are adjacent. If G is regular with degree at least one and S is an independent set of vertices in G , prove that | S | ≤ 1 2 | V ( G ) | . 2. Let G be a simple connected planar graph with dual G * , and suppose G * is simple. Assume G has at least two vertices. Assume G has v vertices, e edges and f faces. Let v r be the number of vertices in G with degree r and let f r be the number of faces of degree r . (a) Prove that 4 f f 3 + 2 e and 4 v v 3 + 2 e . (b) Prove that v 3 + f 3 8. (c) Give an example of a graph where this bound is tight. 3. Suppose the connected graph G has a planar embedding where all faces have even degree. Using the following steps, prove by induction on the number of edges of G that G is bipar- tite. (a) Prove that if G has a bridge
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Unformatted text preview: e , then each component of G-e is bipartite and so G itself is bipartite. (b) Prove that if e is not a bridge then G-e has a planar embedding where all faces have even degree and so G-e is bipartite. (c) Show that if G has no bridge and G-e is bipartite for each edge e , then G is bipartite. 4. Show that if all vertices in G have even degree and E ( G ) is not empty, then G contains a cycle. Using this, prove that if all vertices in G have even degree then we can partition the edge set of G into cycles. 5. Let M 8 be the graph with the integers mod 8 as its vertices, where two vertices x and y are adjacent if x-y ∈ {1,4,7}. Produce a planar embedding of M 8 or prove that it is not planar....
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