부산대학교 도서관

Let $N(n_1,\cdots, n_k)$ denote the number of irreducible coverings by edges of the vertices of a complete $k$-partite graph $K(V_1,\cdots, V_k)$ containing $n_i$ vertices in the class $V_i$ for every $1\leq i\leq k$. In this paper it is proved that $$\multline \sum_{n_1,\cdots, n_k\geq 1}N(n_1,\cdots, n_k)x^{n_1}_1\cdots x^{n_k}_k/(n_1!\cdots n_k!)=\\ \exp\Big (\sum^k_{i=1}(\exp (x_1+\cdots +x_{i-1}+x_{i+1}+\cdots +x_k)x_i-x_i)-\\ \sum_{1\leq iMR1034250 (91g:05008)].