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

TL;DR

This paper proves that the graph formed by removing a complete subgraph from a complete graph can be decomposed into cycles of specified lengths under certain conditions, extending previous cycle decomposition results.

Contribution

It generalizes existing cycle decomposition results to graphs with a hole, allowing arbitrary cycle lengths with specific constraints, broadening the scope of cycle decompositions.

Findings

Decomposition into cycles of arbitrary lengths is possible under given conditions.

Longest cycle length is at most three times the second longest.

Results extend previous uniform cycle decomposition theories.

Abstract

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