The Nordhaus–Gaddum-type inequalities for the Zagreb index and co-index of graphs

Let k≥2k≥2 be an integer, a kk-decomposition  (G1,G2,…,Gk)(G1,G2,…,Gk) of the complete graph KnKn is a partition of its edge set to form kk spanning subgraphs G1,G2,…,GkG1,G2,…,Gk. In this contribution, we investigate the Nordhaus–Gaddum-type inequality of a kk-decomposition of KnKn for the general Zagreb index and a 22-decomposition for the Zagreb co-indices, respectively. The corresponding extremal graphs are characterized.