Pranava K. Jha

L(2,1)-labeling of direct product of paths and cycles

An L (2, 1)-labeling of a graph G is an assignment of labels from {0, 1,....., λ} to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. The λ-number λ(G) of G is the minimum value λ such that G admits an L (2, 1)-labeling. Let G × H denote the direct product of G and H. We compute the λ-numbers for each of C7i × C7j, C11i × C11j × C11k, P4 × Cm, and P5 × Cm. We also show that for n ≥ 6 and m ≥ 7, λ(Pn × Cm) = 6 if and only if m = 7k, k ≥ 1. The results are partially obtained by a computer search.

Optimal L(2, 1)-labeling of strong products of cycles [transmitter frequency assignment]

The L(2, 1)-labeling of a graph is an abstraction of assigning integer frequencies to radio transmitters such that i) transmitters that are one unit of distance apart receive frequencies that differ by at least two, and ii) transmitters that are two units of distance apart receive frequencies that differ by at least one. The least span of frequencies in such a labeling is referred to as the /spl lambda/-number of the graph. It is shown that if k/spl ges/1 and m/sub 0/, ..., m/sub k-1/ are each a multiple of 3/sup k/+2, then /spl lambda/(Cm/sub 0//spl square/.../spl square/Cm/sub k-1/) is equal to the theoretical minimum of 3/sup k/+1, where C/sub i/ denotes the cycle of length i and /spl square/ denotes the strong product of graphs.

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

An L(d,1)-labeling of a graph G is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least d and those at a distance of two receive labels that differ by at least one, where d ≥ 1. Let λ1d(G) denote the least λ such that G admits an L(d,1)-labeling using labels from {0, 1, ..., λ}. We prove that (i) if d ≥ 1, k ≥ 2 and m0, ..., mk-1 are each a multiple of 2k + 2d - 1, then λ1d(Cm0 × ... × Cmk-1) ≤ 2k + 2d - 2, with equality if 1 ≤ d ≤ 2k, and (ii) if d ≥ 1, k ≥ 1 and m0, ..., mk-1 are each a multiple of 2k + 2d - 1, then λ1d (Cm0□ ...□Cmk-1) ≤ 2k + 2d - 2, with equality if 1 ≤ d ≤ 2k.

Perfect r -domination in the Kronecker product of two cycles, with an application to diagonal/toroidal mesh

If r ≥ 1, and m and n are each a multiple of (r + 1)2 + r2, then each isomorphic component of Cm × Cn admits of a vertex partition into (r + 1)2 + r2 perfect r-dominating sets. The result induces a dense packing of Cm × Cn by means of vertexdisjoint subgraphs, each isomorphic to a diagonal array. Areas of applications include efficient resource placement in a diagonal mesh and error-correcting codes.

Perfect r-domination in the Kronecker product of three cycles

If r/spl ges/1, and m/sub 0/, m/sub 1/, and m/sub 2/ are each a multiple of (r+1)/sup 3/+r/sup 3/, then each isomorphic component of the graph C(m/sub 0/)/spl times/C(m/sub 1/)/spl times/C(m/sub 2/) permits a vertex partition into (r+1)/sup 3/+r/sup 3/ perfect r-dominating sets. The result induces a dense packing of C(m/sub 0/)/spl times/C(m/sub 1/)/spl times/C(m/sub 2/) by means of vertex-disjoint subgraphs, each isomorphic to a connected component of P/sub 2r+1//spl times/P/sub 2r+1//spl times/P/sub 2r+1/. Additional results include a general lower bound on r-domination number of a Kronecker product of finitely many cycles. Areas of applications include efficient resource placement in communication networks and error-correcting codes.

Smallest independent dominating sets in Kronecker products of cycles

Abstract Let k⩾2, n=2 k +1, and let m0,…,mk−1 each be a multiple of n. The graph Cm0×⋯×Cmk−1 consists of isomorphic connected components, each of which is (n−1)-regular and admits of a vertex partition into n smallest independent dominating sets. Accordingly, (independent) domination number of each connected component of this graph is equal to (1/n)th of the number of vertices in it.

Hamiltonian Decomposition of the Rectangular Twisted Torus

We show that the 2a\times a rectangular twisted torus introduced by Cámara et al. [5] is edge decomposable into two Hamiltonian cycles. In the process, the 2a × a × a prismatic twisted torus is edge decomposable into three Hamiltonian cycles, and the 2a × a × a prismatic doubly twisted torus admits two edge-disjoint Hamiltonian cycles.

A scheme to construct distance-three codes using Latin squares, with applications to the n -cube

Let B,, denote the set of n-bit binary strings, and let Q, denote the graph of the n-cube where V(Q,> = B, and where two vertices are adjacent iff their Humming distance is exactly one. A subset C of B, is called a code, and the elements of C are referred to as codewords. C is said to be a linear code if the codeword obtained from component-wise sum (modulo 2) of any two elements of C is again in C; otherwise it is a nonlinear code. By a distance-three code is meant a code in which the Hamming distance between any two distinct codewords is at least three. Distance-three codes possess the capability to correct one error and detect two or fewer errors. It is known that if n is of the form 2k 1, then B, admits of a partition into equal-size sets V O,. . . , V, such that each F is a distance-three code and is maximal with respect to this property

A counterexample to Tang and Padubidri's claim about the bisection width of a diagonal mesh

A counterexample is presented to disprove Tang and Padubidris (1994) claim about the bisection width of a diagonal mesh.

Cycle Kronecker products that are representable as optimal circulants

Broere and Hattingh proved that the Kronecker product of two cycles is a circulant if and only if the cycle lengths are coprime. In this paper, we specify which of these Kronecker products are actually optimal circulants. Further, we present their salient characteristics based on their edge decompositions into Hamiltonian cycles. It turns out that certain products thus distinguished have the added property of being tight-optimal, so their average distances are the least among all circulants of the same order and size. A benefit of the present study is that the existing results on the Kronecker product of two cycles may be used to good effect while putting these circulants into practice. The areas of applications include parallel computers, distributed systems and VLSI.