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We prove that the complete graph with a hole $K_{u+w}-K_u$ can be decomposedinto cycles of arbitrary specified lengths provided that the obvious necessaryconditions are satisfied, each cycle has length at most $\min(u,w)$, and thelongest cycle is at most three times as long as the second longest. Thisgeneralises existing results on decomposing the complete graph with a hole intocycles of uniform length, and complements work on decomposing complete graphs,complete multigraphs, and complete multipartite graphs into cycles of arbitraryspecified lengths.

Decomposing $K_{u+w}-K_u$ into cycles of various lengths