On the Eulerian recurrent lengths of complete bipartite graphs and complete graphs

An Eulerian circuit of a graph is a circuit that contains all of the edges of the graph. A graph that has an Eulerian circuit is called an Eulerian graph. The Eulerian recurrent length of an Eulerian graph G is the maximum of the length of a shortest subcycle of an Eulerian circuit of G. In other words, if every Eulerian circuit of an Eulerian graph G has a subcycle of length less than or equal to l, and there is an Eulerian circuit of G that has no subcycle of length less than l, then the Eulerian recurrent length of G is l. The Eulerian recurrent length of graph G is abbreviated to the ERL of G, and denoted by ERL(G). In this paper, the ERL's of complete bipartite graphs are given. Let m and n be positive even integers with m >= n. It is shown that ERL(K-m,K-n) = 2n - 4 if n = m >= 4, and ERL(K-m,K-n) = 2n otherwise. Furthermore, upper and lower bounds on the ERL's of complete graphs are given. It is shown that n - 4 <= ERL(K-n) <= n - 2 holds for every odd integer n greater than or equal to 7.