Decompositions of pseudographs into closed trails of even sizes

We consider a graph L n , with n even, which is a complete graph with an additional loop at each vertex and minus a 1-factor and we prove that it is edge-disjointly decomposable into closed trails of even lengths greater than four, whenever these lengths sum up to the size of the graph L n . We also show that this statement remains true if we remove from L n two loops attached to nonadjacent vertices. Consequently, we improve P. Wittmann’s result on the upper bound of the irregular coloring number c ( G ) of a 2-regular graph G of size n , by determining that this number is, with a discrepancy of at most one, equal to ⌈ 2 n ⌉ if all components of G have even orders.