Path decomposition of graphs with given path length

A path decomposition of a graph G is a list of paths such that each edge appears in exactly one path in the list. G is said to admit a {P l}-decomposition if G can be decomposed into some copies of P l , where P l is a path of length l -1. Similarly, G is said to admit a {P l, P k }-decomposition if G can be decomposed into some copies of P l or P k . An k-cycle, denoted by C k , is a cycle with k vertices. An odd tree is a tree of which all vertices have odd degree. In this paper, it is shown that a connected graph G admits a {P 3, P 4}-decomposition if and only if G is neither a 3-cycle nor an odd tree. This result includes the related result of Yan, Xu and Mutu. Moreover, two polynomial algorithms are given to find {P 3}-decomposition and {P 3, P 4}-decomposition of graphs, respectively. Hence, {P 3}-decomposition problem and {P 3, P 4}-decomposition problem of graphs are solved completely. © 2006 Springer-Verlag Berlin Heidelberg.