For MSO$_2$-expressible problems like Edge Dominating Set or Hamiltonian Cycle, it was open for a long time whether there is an algorithm which given a clique-width $k$-expression of an $n$-vertex graph runs in time $f(k) \cdot n^{\mathcal{O}(1)}$ for some function $f$. Recently, Fomin et al. (SIAM. J. Computing, 2014) presented several lower bounds; for instance, there are no $f(k)\cdot n^{o(k)}$-time algorithms for Edge Dominating Set and for Hamiltonian Cycle unless the Exponential Time Hypothesis (ETH) fails. They also provided an algorithm running in time $n^{\mathcal{O}(k)}$ for Edge Dominating Set, but left open whether Hamiltonian Cycle can be solved in time $n^{\mathcal{O}(k)}$. In this paper, we prove that Hamiltonian Cycle can be solved in time $n^{\mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{\mathcal{O}(k^2)}$ by Espelage et al. (WG 2001). We present a general technique of representative sets using two-edge colored multigraphs on $k$ vertices. The essential idea behind is that for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternatively can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on $k$ vertices with at most $n$ edges. We can apply this technique to other problems such as $q$-Cycle Covering or Directed Hamiltonian Cycle as well.