Proof: Ore's Theorem for Hamiltonian Graphs | Sufficient Condition for Hamilton Graphs, Graph Theory

What is Ore's Theorem for Hamiltonian graphs and how do we prove it? Ore's Theorem gives us a sufficient condition for a graph to have a Hamiltonian cycle and therefore be a Hamiltonian or Hamilton graph. The theorem tells us that if, in a graph with order n greater than or equal to 3, the degree sum of any pair of non-adjacent vertices is greater than or equal to n, then the graph is Hamiltonian. Put very simply, this tells us if a graph has so many edges that it fits the aforementioned conditioned, then it must be Hamiltonian.

Notice that the condition doesn't specify directly that the graph must be connected, but if a graph fulfills the condition it will inevitably be connected, and this can be easily proven. Our proof will use an argument by contradiction, a maximally-non-Hamiltonian supergraph, a Hamilton path, and more! It's a fun ride!

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