Given an eulerian graph G and an Euler tour T of G, the girth of T, denoted by g(T), is the minimum integer k such that some segment of k+1 consecutive vertices of T is a cycle of length k in G. Let g E (G)= maxg(T) where the maximum is taken over all Euler tours of G. We prove that g E (K 2n,2n )=4n-4 and 2n-3=g E (K 2n+1)=2n-1 for any n=2. We also show that g E (K 7)=4. We use these results to prove the following: 1)The graph K 2n,2n can be decomposed into edge disjoint paths of length k if and only if k=4n-1 and the number of edges in K 2n,2n is divisible by k. 2)The graph K 2n+1 can be decomposed into edge disjoint paths of length k if and only if k=2n and the number edges in K 2n+1 is divisible by k. © Springer-Verlag 2005.

We consider the problem of PMk+1-decomposition of a simple eulerian graph G, that is, decomposition of G into edge disjoint paths of length k. We show that the problem of deciding whether there exists a Pk+1 - decomposition of an eulerian simple graph is NP-complete, for every k = 3. However we find some new classes of graphs where the problem of P4-decomposition can be solved polynomially. We show that an eulerian simple graph G on 3m = 6 edges admits a P4-decomposition if G has no cut vertex v such that exactly one of the components in the graph G - ? has two vertices. In particular, this implies that a 2-connected eulerian simple graph G on 3m = 6 edges admits a P4 -decomposition. © 2003.