On the complete bipartite decomposition of planar graphs

Given a graph G, a complete bipartite decomposition of G is a partition pi of E(G) into disjoint sets E(H-i) such that each subgraph H-i induced by E(H-i) is an edge-disjoint union of complete bipartite subgraphs of G. G's complete bipartite capacity c(G) is the minimum cardinality of pi over all of complete bipartite decompositions, and its complete bipartite valency theta(G) is the minimum, over all complete bipartite decompositions pi, of the maximum number of elements in pi incident with a vertex. A star decomposition, the star capacity (or conventionally, star arboricity) st(G) and the star valency theta(s)(G) of a graph G are defined analogously. In this article, we establish the sharp upper bounds on theta(G), c(G), st(G) and theta(s)(G) for outerplanar graphs and on theta(G), theta(s)(G) and c(G) for planar graphs. We also find the characteristic of graphs with small value of theta(s)(G) and furthermore the classification of planar graphs in terms of theta(s)(G).