I tried drawing all the possible non-isomorphic 4-regular graphs with 8 vertices and I noticed that all of them have a cycle of length 8. I tried proving it but I didn't know how to start. How would you prove that a specific graph must have a cycle of length x?
How to prove that a 4-regular graph with 8 vertices has a cycle of length 8
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graph-theory
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0So have you followed any of those links? Anything to say about the question? – 2017-01-14
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0Earth to Bhargavi, come in, please. – 2017-01-16
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0Hey Gerry! Thanks for the reply! The question was a simple observation I made while solving an assignment question. I'm new to graph theory and I guess I'm not equipped with the theorems needed to understand it. But the links definitely helped! – 2017-01-19
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0Good. You have the option of "accepting" then answer by clicking in the check mark next to it. – 2017-01-19
1 Answers
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Dirac's Theorem from 1952 says that if $n\ge3$ then a simple graph with $n$ vertices is Hamiltonian if every vertex has degree $n/2$ or greater. See also Ore's Theorem from 1960.
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0See also http://math.stackexchange.com/questions/59159/understanding-the-proof-of-diracs-theorem-regarding-graph-connectivity and http://math.stackexchange.com/questions/624766/clarifying-diracs-theorem and http://math.stackexchange.com/questions/1602222/diracs-theorem-not-work and http://math.stackexchange.com/questions/1701605/prove-diracs-theorem-by-induction-on-the-number-of-vertices and http://math.stackexchange.com/questions/690491/pigeonhole-principle-to-prove-a-hamiltonian-graph/690522 and probably quite a few others. – 2017-01-13