Decomposition of the Johnson Graphs into Graph-Pairs of Order 4

A graph-pair of order t is a pair of graphs G and H on t non-isolated vertices for which \(G \cup H \cong K_t\) for some integer \(t \ge 4\) . The Johnson graph J(v, n) is the graph whose vertices are the n-element subsets of a v-element set, and two vertices are adjacent if the intersection of the corresponding subsets contains \(n-1\) elements. We show necessary and sufficient conditions for J(v, 2) to admit a decomposition into graph-pairs of order 4.