Directed tree decompositions of Cayley digraphs with word-degenerate connection sets

For a tree T , the graph X is T -decomposable if there exists a partition of the edge set of X into isomorphic copies of T . In 1963, Ringel conjectured that K2m+1 can be decomposed by any tree with m edges. Graham and Häggkvist conjectured more generally that every 2m-regular graph can be decomposed by any tree with m edges. Fink showed in 1994 that for any directed tree T with m arcs, the directed Cayley graph DCay(G;S) is T -decomposable if S is a minimal generating set of G with m elements. Building upon that technique, this paper presents an enlarged family of directed Cayley graphs that are decomposable into directed trees. In particular, a subset S of a finite group G is defined to be (k, t)-word degenerate if S contains exactly t elements, s1, . . . , st, such that for each i ∈ {1, . . . , t}, si can be expressed as a product of fewer than k distinct elements from S − {si} or their inverses. It is proved that if S is any (k, t)-word degenerate m-subset of a group G, and T is any directed tree having m arcs and a minimal spanning star forest F , then the directed Cayley graph DCay(G;S) is T -decomposable whenever k ≥ diam(T ) ≥ 3, and t ≤ |E(F )|. When diam(T ) = 2, additional restrictions are required. The main result of Fink and other results are obtained as immediate corollaries.