Euler Tours of Maximum Girth in K_{2 n +1} and K_{2 n ,2 n}

Abstract.

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_{2 n ,2 n} )=4n−4 and 2n−3≤g^E(K_{2 n +1})≤2n−1 for any n≥2. We also show that g^E(K₇)=4. We use these results to prove the following:

1)The graph K_{2 n ,2 n} can be decomposed into edge disjoint paths of length k if and only if k≤4n−1 and the number of edges in K_{2 n ,2 n} is divisible by k.

2)The graph K_{2 n +1} can be decomposed into edge disjoint paths of length k if and only if k≤2n and the number edges in K_{2 n +1} is divisible by k.