Abstract
AbstractWe study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic ‐vertex graph with and a random ‐regular graph , for . When is a random 2‐regular graph, we prove that a.a.s. is pancyclic for all , and also extend our result to a range of sublinear degrees. When is a random 1‐regular graph, we prove that a.a.s. is pancyclic for all , and this result is best possible. Furthermore, we show that this bound on is only needed when is “far” from containing a perfect matching, as otherwise we can show results analogous to those for random 2‐regular graphs. Our proofs provide polynomial‐time algorithms to find cycles of any length.
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