2012•Journal of Combinatorial DesignsRequires access

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

Emre Kolotoğlu

Abstract

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a -design of order n. The existence problem of -designs has been completely solved for the graphs for , for , K2, 3 and K3, 3. In this paper, I prove that for all , if there exists a -design of order N, then there exists a -design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.

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Key concepts: Mathematics, Combinatorics, Bipartite graph, Cograph, Robertson–Seymour theorem, Complete bipartite graph, Discrete mathematics, Disjoint sets