An exact bound on the number of proper $3$-edge-colorings of a connected cubic graph

In this paper, the problem of finding an upper bound on the number of proper edge $3$-colorings of a connected cubic graph with $2n$ vertices is explored. For this purpose, we extended Karpov's method which allowed to obtain a weaker result earlier. The bounds $2^n+8$ for even $n$ and $2^n+4$ for odd $n$ was proved. We have found the unique series of graphs on which these bounds are attained. Thus for, in this problem the exact upper bound has been found and proved.