Decompositions of pseudographs into closed trails of even sizes

We consider a graph LnLn, with nn 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 LnLn. We also show that this statement remains true if we remove from LnLn two loops attached to nonadjacent vertices. Consequently, we improve P. Wittmann’s result on the upper bound of the irregular coloring number c(G)c(G) of a 2-regular graph GG of size nn, by determining that this number is, with a discrepancy of at most one, equal to ⌈2n⌉ if all components of GG have even orders.