Unformatted text preview: e , then each component of Ge is bipartite and so G itself is bipartite. (b) Prove that if e is not a bridge then Ge has a planar embedding where all faces have even degree and so Ge is bipartite. (c) Show that if G has no bridge and Ge 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 xy ∈ {1,4,7}. Produce a planar embedding of M 8 or prove that it is not planar....
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 Spring '09
 M.PEI
 Math, Graph Theory, Planar graph, vertices, planar embedding

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