A constrained path decomposition of cubic graphs and the path number of cacti
Kotzig (1957) proved that a cubic graph has a perfect matching if and only if it has a 3-path decomposition (that is, a partition of the edge set into paths of length 3). This result was generalized by Jaeger, Payan, and Kouider (1983), who proved that a (2k + l)-regular graph with a perfect matching can be decomposed into bistars. (A bistar is a graph obtained from two disjoint stars by joining their centers with an edge.) In another direction, Heinrich, Liu and Yu (1999) proved that a 3m-regular graph G admits a balanced 3-path decomposition if and only if G contains an m-factor.