Decomposing various graphs into short even-length cycles

TL;DR

This paper proves conditions under which complete bipartite graphs can be decomposed into specified even-length cycles and applies these results to incomplete cycle systems and multipartite graphs.

Contribution

It establishes new decomposition theorems for bipartite and multipartite graphs into even cycles with specific length constraints.

Findings

Complete bipartite graphs can be decomposed into cycles of arbitrary specified lengths under certain conditions.

Results extend to incomplete even cycle systems with a hole.

Decompositions of complete multipartite graphs into uniform even cycles are achieved.

Abstract

We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the longest cycle is at most three times as long as the second longest. We then use this result to obtain results on incomplete even cycle systems with a hole and on decompositions of complete multipartite graphs into cycles of uniform even length.