Enumeration of labeled Eulerian pentacyclic graphs

An Euler graph is a connected graph in which all degrees of vertices are even numbers. A pentacyclic graph is a connected graph with $n$ vertices and $n + 4$ edges. We obtain an explicit formula for the number of labeled Euler pentacyclic graphs with a given number of vertices, and found the corresponding asymptotics for the number of such graphs with a large number of vertices. We prove that, given a uniform probability distribution, the probability that a labeled pentacyclic Euler graph is a block (cactus) is asymptotically $53/272$ ($63/272$), respectively.