QUESTION 5 Suppose that G is a plane graph that has 15 edges in the boundary of its exterior region and all the other regions of G contain 4, 6, or 8 regions in their boundary. Use Grinberg's Theorem to show that G cannot contain a Hamilton circuit.
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Grinberg's Theorem states that if a plane graph G has n vertices and every region of G contains at least k edges in its boundary, then G cannot contain a Hamilton circuit if n > 2k. Show more…
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Show that if $u$ and $v$ are nonadjacent vertices in a graph $G$ with $n$ vertices and $\operatorname{deg}(u)+\operatorname{deg}(v) \geq n,$ then $G$ has a Hamilton circuit if and only if $G+\{u, v\}$ has a Hamilton circuit.
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Theorem 3 (Grinberg; 1968): Suppose a planar graph G has a Hamilton circuit H. Let G be drawn with any planar depiction, and let r denote the number of regions inside the Hamilton circuit bounded by edges in this depiction. Let r' be the number of regions outside the circuit bounded by edges. Then the numbers r and r' satisfy the equation (r - 2)(r + r') = 0.
Sri K.
Grinberg's theorem states that a planar graph that has a Hamilton circuit satisfies the formula: ∑i (i - 2)(ri - ri') = 0. Here, ri resp. ri' is the number of regions that have i sides and lay inside resp. outside the Hamilton circuit. Use Grinberg's theorem to show that the graph shown below has no Hamilton circuit.
Adi S.
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