On the dimension of vertex labeling of k-uniform dcsl of k-uniform caterpillar
A distance compatible set labeling (dcsl) of a connected graph G is an injective set assignment f : V(G) → 2^X, X being a nonempty ground set, such that the corresponding induced function f^ :E(G) → 2^X{} given by f^(uv)= f(u)f(v) satisfies f^(uv) = k_(u,v)^fd_G(u,v) for every pair of distinct vertices u, v V(G), where d_G(u,v) denotes the path distance between u and v and k_(u,v)^f is a constant, not necessarily an integer. A dcsl f of G is k-uniform if all the constant of proportionality with respect to f are equal to k, and if G admits such a dcsl then G is called a k-uniform dcsl graph. The k-uniform dcsl index of a graph G, denoted by δ_k(G) is the minimum of the cardinalities of X, as X varies over all k-uniform dcsl-sets of G. A linear extension L of a partial order P = (P, ) is a linear order on the elements of P, such that x y in P implies x y in L , for all x, y P. The dimension of a poset P , denoted by dim(P ), is the minimum number of linear extensions on P whose intersection is `'. In this paper we prove that dim(F ) ≤ δ_k(P^+k_n), where F is the range of a k-uniform dcsl of the k-uniform caterpillar, denoted by P^+k_n (n≥ 1, k≥ 1) on `n(k+1)' vertices. , number=1, journal=Carpathian Mathematical Publications, author=Nageswara Rao, K. and Germina, K.A. and Shaini, P., year=2016, month=Jun., pages=134–149