Hamiltonicity of graphs perturbed by a random regular graph

We study Hamiltonicity and pancyclicity in the graph obtained as union of a deterministic n $$ -vertex H with δ ( ) ≥ α \delta (H)\ge \alpha random d -regular G , for ∈ { 1 2 } d\in \left\{1,2\right\} . When is 2-regular graph, we prove that a.a.s. ∪ H\cup pancyclic all 0 ] \in \left(0,1\right] also extend our result to range sublinear degrees. 1-regular − \left(\sqrt{2}-1,1\right] this best possible. Furthermore, show bound on (H) only needed when “far” from containing perfect matching, otherwise can results analogous those graphs. Our proofs provide polynomial-time algorithms find cycles any length.