Constructions of Quad-Rooted Double Covers

A collection of spanning subgraphs of Kn is called an orthogonal double cover if (i) every edge of Kn belongs to exactly two of the Gi’s and (ii) any two distinct Gi’s intersect in exactly one edge. Chung and West [3] conjectured that there exists an orthogonal double cover of Kn for all n, in which each Gi has maximum degree 2, and proved this result for n in six of the residue classes modulo 12. In [6], Gronau, Mullin and Schellenberg solved the conjecture. In addition to solving the conjecture, they went on to consider a problem for n ≡ 5 mod 6 such that each spanning subgraph Gi consists of the vertex-disjoint union of an isolated vertex, a quadrilateral, and triangles. They proved that for any n ≡ 2 mod 3 and n∉ {8, 11, 38, 41, 44, 47, 50, 53, 59, 62, 71, 83, 86, 89, 95, 101, 107, 113, 122, 131, 143, 146, 149, 158, 164, 167, 173, 176, 179, 218, 242, 248, 287}, there exists a quad-rooted double cover of order n. In this note, we improve their result by showing that such designs exist for any n ≡ 2 mod 3 and n∉ {8, 11, 38, 41, 44, 50, 53, 62, 71}.