The Spectrum Problem for the 4-Uniform 4-Colorable 3-Cycles with Maximum Degree 2

The complete t-uniform hypergraph of order v, denoted \(K^{(t)}_v\) , has a set V  with v elements as its vertex set and the set of all t-element subsets of V  as its edge set. For the purposes of this work, we define a 4-uniform 3-cycle of maximum degree 2 to be any 4-uniform hypergraph of maximum degree 2 that can be obtained by adding two vertices to each of the three edges in \(K^{(2)}_3\) . There are five such 4-uniform hypergraphs up to isomorphism. Two of them have chromatic number 4. We give necessary and sufficient conditions for the existence of a decomposition of the complete 4-uniform hypergraph of order v into these 4-colorable 3-cycles.