On L(h, K)-labellings Of A Type Of Cartesian Products Of Graphs

For positive integers h, k, an L(h,k)-labelling of a graph G is a function f : V(G)→{0,1,2,..., m} such that if d(u, v) = 2. The span of an L(h,k)-labelling is the difference between the largest and the smallest label. Theλ_h,k-number of G is the minimum span over the L(h, k)-labellings of G. In this paper, We investigate L(h,k)-labelling of a class of Cartesian products of graphs, X_△□P₂, Y_△□P₂, Z_△□P₂, W_△□P₂, where X_â–³is the star K_0,â–³,Y_â–³is the graph obtained by attaching a pendant edge to a pendant vertex of the star K_1,â–³, Z_â–³by attaching two pendant edges to a pendant vertex of the star K_1,â–³, W_â–³by attaching a path P_d of length d - 1 to a pendant vertex of K_1,â–³ ford≥3.1. The detailed proofs of these results are presented in the subsequent sections. In all cases, upper bounds onλ_h,k(â–³) are proved by constructing an L(h,k)-labelling for X_△□P₂ .Lower bounds onλ_h,k(â–³) are proved by showing that a star of maximum degree A, for which every L(h, k)-labelling for the Cartesian product of the star and P₂ has span at least equal to some given integer.2. On the third section, We derive theλ_h,k-number of X_△□P₂ (respectively, for all values ofâ–³, h and k.3. On the fourth section,Under certain conditions, we also determine theλ_h,k-number of Z_△□P₂ and W_△□P₂.4. We have not,completely established the minimum span for Z_△□P₂and W_△□P₂ for△≥3. So we list the unresolved problems in the following for further research.