The representing formula of N(G,k)

In this paper, by combinatorial methods, the authors derive the representing formula between the number N(G,k) of S(n) = {Ki : 1 ≤ i ≤n}-factors with exactly k components and coefficients of the chromatic poly nomial of ¯G, and the representing formula between the number A(G) of allS(n)-factors and coefficients of the chromatic polynomial of ¯G, present classes of graphs N(G,k) and A(G), and give combinatorial identities related toN(G,k).Finally, we solve the counting formula of Kd−factors in any complete d-partite graphs,the counting theorem of Kl − factors in any graph Gwithout any Kl+1 subgraph, the counting formula of the number of covers on shortest circles, as well as the explicit formula µ(Kd n) of the mean colornumbers of Kdn.