Problem 1a)show me a planar graph G with 6 vertices and 8 edges, such that G is bipartiteb)show me a...

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Problem 1a)show me a planar graph G with 6 vertices and 8 edges, such that G is bipartiteb)show me a planar H with 6 vertices and 8 edges such that H is not bipartite
Problem 2You are given the degree sequence 4,4,3,3,2,2,. Find as many non-isomorphic graphs as you can with this degree sequence. Be sure to tell me how you know they are different.Problem 3Prove the following: if a graph G has n vertices and the sum of the vertex degrees of G is strictly greater than 2n, the G must contain at least two different circuits. "Different" means that they are not identical they have at least one edge which is different.

David · Accepted Answer

Problem 1 
a)show me a planar graph G with 6 vertices and 8 edges, such that G is bipartite
Kuratowski’s theorem tells us that, if we can find a sub-graph in any graph that is homeomorphic to K5 or K 3,3, 
then the graph is not planar, it is  not possible for the edges to be redrawn such that they are none overlapping.
None of the (6C3)=20 arrangement of 3 edges are complete. The graph is not complete of higher order. 
The graph is planar.  We have only even cycles.
b)show me a planar H with 6 vertices and 8 edges such that H is not bipartite
Planar though it may be, the graph above is non-bipatite.

Problem 1a)show me a planar graph G with 6 vertices and 8 edges, such that G is bipartiteb)show me a planar H with 6 vertices and 8 edges such that H is not bipartite Problem 2You are given the degree...

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