On the Roman domination problem of some Johnson graphs

A Roman domination function (RDF) on a graph G with a set of vertices V = V(G) is a function f : V ? {0, 1, 2} which satisfies the condition that each vertex v ? V such that f (v) = 0 is adjacent to at least one vertex u such that f (u) = 2. The minimum weight value of an RDF on graph G is called the Roman domination number (RDN) of G and it is denoted by ?R(G). An RDF for which ?R(G) is achieved is called a ?R(G)-function. This paper considers Roman domination problem for Johnson graphs Jn,2 and Jn,3. For Jn,2, n ? 4 it is proved that ?R(Jn,2) = n ? 1. New lower and upper bounds for Jn,3, n ? 6 are derived using results on the minimal coverings of pairs by triples. These bounds quadratically depend on dimension n.