A (k,k) bipartite clique is a bipartite graph with k vertices in each side, where each vertex on the left and each vertex on the right are adjacent by an edge.Suppose you are told that a (3,3)-bipartite clique is not a planar graph. Is a (4,4)-bipartite clique planar?
Question
A (k,k) bipartite clique is a bipartite graph with k vertices in each side, where each vertex on the left and each vertex on the right are adjacent by an edge.Suppose you are told that a (3,3)-bipartite clique is not a planar graph. Is a (4,4)-bipartite clique planar?
Solution
No, a (4,4)-bipartite clique is not a planar graph. This is because a planar graph cannot have a subgraph that is homeomorphic to K5 (complete graph on 5 vertices) or K3,3 (complete bipartite graph on 6 vertices with partition sizes 3,3). A (4,4)-bipartite clique is a K4,4 graph, which contains K3,3 as a subgraph, so it cannot be planar.
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