The minimum span of L(2,1)-labelings of generalized flowers

Given a positive integer d, an L(d, 1)-labeling of a graph G is an assignment of nonnegative integers to its vertices such that adjacent vertices must receive integers at least d apart, and vertices at distance two must receive integers at least one apart. The lambda(d)-number of G is the minimum k so that G has an L(d, 1)-labeling using labels in {0, 1, ... , k}. informally, an amalgamation of two disjoint graphs G(1) and G(2) along a fixed graph G(0) is the simple graph obtained by identifying the vertices of two induced subgraphs isomorphic to G(0), one in G(1) and the other in G(2). A flower is an amalgamation of two or more cycles along a single vertex. We provide the exact lambda(2)-number of a generalized flower which is the Cartesian product of a path P-n and a flower, or equivalently, an amalgamation of cylindrical rectangular grids along a certain P. In the process, we provide general upper bounds for the lambda(d)-number of the Cartesian product of P and any graph G, using circular L(d+1, 1)-labelings of G where the labels (0, 1, ..., k) are arranged sequentially in a circle and the distance between two labels is the shortest distance on the circle. (C) 2014 Elsevier B.V. All rights reserved.