Decomposition of complete graphs into 6-stars and into 10-stars

Abstract

A q-star is a connected graph with q edges and in which every vertex but one has valency 1. This paper concerns the question of which particular complete graphs can be decomposed into q-stars that have pairwise disjoint edge-sets for the values of q, 6 and 10. It is shown that the complete graphs on m vertices can be decomposed into 6-stars if and only if m is greater than or equal to 12 and m ≡ 0,1,4,9 (mod 12). It is also shown that the complete graphs on m vertices can be decomposed into 10-stars if and only if m is greater than or equal to 20 and m ≡ 0,1,5,16 (mod 20).