Optimal L(2,1)-labeling of Cartesian products of cycles, with an application to independent domination

Abstract

The (2 1)-labeling of a graph is an abstraction of the problem of assigning (integer) frequencies to radio transmitters, such that transmitters that are "close", receive different frequencies, and those that are "very close" receive frequencies that are further apart. The least span of frequencies in such a labeling is referred to as the -number of the graph. Let be odd 5, = ( 3) 2 and let each be a multiple of . It is shown that ( ) is equal to the theoretical minimum of 1, where denotes a cycle of length and " " denotes the Cartesian product of graphs. The scheme works for a vertex partition of into smallest (independent) dominating sets.

Optimal L(d,1)-labelings of certain direct products of cycles and Cartesian products of cycles

“…. , m k−1 are each a multiple of 2k + 3, then 2 1 (C m 0 · · · C m k−1 ) = 2k + 2 [7]. These results follow from Theorem 4 for d = 1 and d = 2, respectively.…”

“…Analogous result is known with respect to 2 1 -numbering of the strong products of cycles [8]. For results with respect to Cartesian products, see [2,7,10,11,14,15]. The following fact will be useful in the sequel.…”

“…Georges and Mauro [1] later presented a generalization of the concept. The topic has since been an object of extensive research [1][2][3][4][7][8][9][10][11][12]14,15].…”

L(2,1)-labelings of Cartesian products of two cycles

“…The -number of the Cartesian products of graphs was investigated in [11,12,[14][15][16][17]24]. The Cartesian product of two graphs G and H (or simply product), denoted by G H , is defined as the graph with vertex set given by the Cartesian product of the vertex set of G and the vertex set of H, where two vertices (u, v) and (w, z) are adjacent if and only if either [u, w are adjacent in G and v = z] or [v, z are adjacent in H and u = w].…”

The L(2,1) -Labelings on the Homomorphic Product of two Graphs

“…L -labeling has been extensively studied in recent past by many researchers [see 2,5,9,10,12,13,20]. The common trend in most of the research paper is either to determine the value of (2,1) L -labeling number or to suggest bounds for particular classes of graphs.…”