The Strong Equitable Vertex 1-Arboricity of Complete Bipartite Graphs and Balanced Complete k-Partite Graphs

An equitable k-coloring of a graph G is proper such that the sizes any two color classes differ by at most one. (q,r)-tree-coloring an q-coloring subgraph induced each class forest maximum degree r. Let strong vertex r-arboricity G, denoted var≡(G), be minimum p has for every q≥p. The values va1≡(Kn,n) were investigated Tao and Lin Wu, Zhang, Li where exact found in some special cases. In this paper, we extend their results giving all process, introduce new function related to coloring obtain more general result determining value va1≡(Km,n) va1≡(G) balanced complete k-partite Kn,…,n. Both bipartite graphs Km,n Kn,…,n are symmetry several aspects also studied broadly. For other aspect symmetry, definition graphs, specific case number colors divides vertices graph, can say graph.