On minimum index Stanton 4-cycle designs (Q2089529)

A Stanton 4-cycle is a multigraph whose underlying simple graph is a 4-cycle and where each edge multiplicity in the set \(\{1,2,3,4\}\) is represented. There are three such multigraphs, up to isomorphism, denoted by \(G_1\), \(G_2\), and \(G_3\) (see Fig. 1 on page 100). For each integer \(n\geq 4\), the authors consider the problem of determining the minimum \(\lambda\) that guarantees the existence of a decomposition of \(^\lambda K_n\) (the \(\lambda\)-fold complete graph on \(n\) vertices) into copies of \(G_i\), \(i=1,2,3\). The problem is completely solved in [\textit{R. C. Bunge} et al., Congr. Numerantium 215, 7--16 (2013; Zbl 1290.05049)] when \(i=1\). In this paper, using specific vertex labelings and special constructions for some small cases, the authors completely solve the above problem for the remaining two Stanton \(4\)-cycles. For the entire collection see [Zbl 1495.05003].