Let $S(G^sigma)$ be the skew-adjacency matrix of an oriented graph $G^sigma$. The skew energy of $G^sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^sigma)$. In this paper, we first deduce an integral formula for
the skew energy of an oriented graph. Then we determine all oriented graphs with minimal skew energy among all connected oriented graphs on $n$ vertices with $m (nle m < 2(n-2))$ arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi {it et al.} [G. Caporossi, D. Cvetkovi$acute{c}$, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.]
The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoi
nt monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $ngeq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$
