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Show that any connected triangle free planar graph has at

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A simple graph is triangle-free when it has no simple cycle of length three.

(a) Prove for any connected triangle-free planar graph with v>2 vertices and e edges. e < 2v - 4.

Hint: Similar to the proof that e < 3v - 6.

(b) Show that any connected triangle-free planar graph has at least one vertex of degree three orless.

(c) Prove by induction on the number of vertices that any connected triangle-free planar graph is 4-colorable. 

Hint: use part b

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Use f denote the number of faces in the graph, s1 to denote the number of sides on face i, and e to denote the number of edges in the graph. Now if we sum the sis we are going to get

Reference no: EM131177508

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