On Regular Graphs Optimally Labeled with a Condition at Distance Two

For positive integers j ≥ k, the λj,k-number of graph G is the smallest span among all integer labelings of V (G) such that vertices at distance two receive labels which differ by at least k and adjacent vertices receive labels which differ by at least j. We prove that the λj,k-number of any r-regular graph is no less than the λj,k-number of the infinite r-regular tree T∞(r). Defining an r-regular graph G to be (j, k, r)-optimal if and only if λj,k(G) = λj,k(T∞(r)), we establish the equivalence between (j, k, r)-optimal graphs and r-regular bipartite graphs with a certain edge coloring property for the case j k > r. The structure of r-regular optimal graphs for j k ≤ r is investigated, with special attention to j k = 1, 2. For the latter, we establish that a (2, 1, r)-optimal graph, through a series of edge transformations, has a canonical form. Finally, we apply our results on optimality to the derivation of the λj,k-numbers of prisms.