Ore’s theorem. A connected graph with N vertices is said tosatisfy Ore’s condition if deg (X) + deg (Y) ≥ N for everypair of vertices X and Y of the graph. Ore’s theorem statesthat if a graph satisfies Ore’s condition, then it has a Hamiltoncircuit.
(a) Explain why the complete bipartite graph Km, n (seeExercise 71) satisfies Ore’s condition.
(b) Explain why for m ≠ n, the complete bipartite graphKm, n (see Exercise 71) does not satisfy Ore’s condition.
(c) Ore’s condition is sufficient to guarantee that a connected graph has a Hamilton circuit but is not a necessary condition. Give an example of a graph thathas a Hamilton circuit but does not satisfy Ore’s condition.
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