Disconnected selfcomplementary factors of almost complete tripartite graphs

Abstract

A complete tripartite graph without one edge, K̃_{m1 ,m2 ,m3} , is called an almost complete tripartite graph. A graph K̃_{m1 ,m2 ,m3} that can be decomposed into two isomorphic factors is called halvable. It is proved that an almost complete tripartite graph is halvable into disconnected factors without isolated vertices if an only if it is a graph K̃_{1, 2m+1 ,2p} and the "missing" (i.e., deleted) edge has the endvertices in the odd parts. It is also shown that the factors have always two components: one component is isomorphic to a star K_{1 ,p} and the other to a graph K_{1 ,2m ,p} - K_{1 ,m}. For factors with isolated vertices it is proved that they have just one non-trivial component and all isolated vertices belong to the same part.