Homework Answers
NO , IT IS NOT TRUE THAT EVERY NON-PLANAR GRAPH REQUIRES MORE THAN 4 COLORS.
EXAMPLE - K3,3 IS NON-PLANAR AND BIPARTITE GRAPH WHICH IS 2 COLORABLE.
K3,3 is 2 colorable because every bipartite graph is 2 colorable as the vertex set of bipartite graph can be divided into 2 sets say (X and Y) such that there is no edge between vertices of X and no edge between vertices of Y.
so none of the vertex of X is adjacent to other vertex of X , so all the vertices of X can be assigned one color say red.
similarly , none of the vertex of Y is adjacent to other vertex of Y , so all the vertices of Y can be assigned one color which is different from the previous color say blue.
so the bipartite graphs are 2 colorable and hence K3,3 is 2 colorable which is non-planar
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Can some one please help me with this two
questions.
Thank you!
fact that every planar graph has a vertex of degree s 5 to give a simple induction proof that every planar graph can be 6-colored. What can be said about the chromatic number of a graph that has Kn as a subgraph? Justify your answer
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