Constructions of Quad-Rooted Double Covers

Abstract

A collection of spanning subgraphs of K _n is called an orthogonal double cover if (i) every edge of K _n belongs to exactly two of the G _i ’s and (ii) any two distinct G _i ’s intersect in exactly one edge. Chung and West [3] conjectured that there exists an orthogonal double cover of K _n for all n, in which each G _i 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 G _i 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}.