Distance-two labellings of Hamming graphs

Discussiones Mathematicae Graph Theory

An L(2, 1)-labeling of a graph Γ is an assignment of non-negative integers to the vertices such that adjacent vertices receive labels that differ by at least 2, and those at a distance of two receive labels that differ by at least one. Let λ 1 2 (Γ) denote the least λ such that Γ admits an L(2, 1)-labeling using labels from {0, 1,. .. , λ}. A Cayley graph of group G is called a circulant graph of order n, if G = Z n. In this paper initially we investigate the upper bound for the span of the L(2, 1)-labeling for Cayley graphs on cyclic groups with "large" connection sets. Then we extend our observation and find the span of L(2, 1)-labeling for any circulants of order n.

European Journal of Combinatorics, 2003

For given positive integers j ≥ k, an L(j, k)-labeling of a graph G is a function f : V (G) → {0, 1, 2,. . .} such that | f (u) − f (v)| ≥ j when d G (u, v) = 1 and | f (u) − f (v)| ≥ k when d G (u, v) = 2. The L(j, k)-labeling number λ j,k (G) of G is defined as the minimum m such that there is an L(j, k)-labeling f of G with f (V (G)) ⊆ {0, 1, 2,. .. , m}. For a graph G of maximum degree ∆ ≥ 1 it is the case that λ j,k (G) ≥ j + (∆ − 1)k. The purpose of this paper is to study the structures of graphs G with maximum degree ∆ ≥ 1 and λ j,k (G) = j + (∆ − 1)k.

Journal of Combinatorial Optimization, 2012

A k-L(2, 1)-labelling of a graph G is a mapping f : V (G) → {0, 1, 2,. .. , k} such that |f (u) − f (v)| ≥ 2 if uv ∈ E(G) and f (u) = f (v) if u, v are distance two apart. The smallest positive integer k such that G admits a k-L(2, 1)-labelling is called the λ-number of G. In this paper we study this quantity for cubic Cayley graphs (other than the prism graphs) on dihedral groups, which are called brick product graphs or honeycomb toroidal graphs. We prove that the λ-number of such a graph is between 5 and 7, and moreover we give a characterisation of such graphs with λ-number 5.

IEEE Transactions on Circuits and Systems II: Express Briefs, 2000

The concept of L(2,1)-labeling in the graph came into existence with the solution of the frequency assignment problem. In fact, in this problem a frequency in the form of nonnegative integers is to assign to every radio or TV transmitters located at different places such that communication does not interfere. This frequency assignment problem can be modeled with vertex labeling of graphs. An L(2,1)-labeling (or distance two labeling) of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(u)-f(v)|≥2 ; d(u,v)=1 and |f(u)-f(v)|≥1 ; d(u,v)=2, where d(u,v) denotes distance between u and v in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling withmax {f(v):v ϵ V(G) }=k. This paper considers the L(2,1)-labeling number for the homomorphic product of two graphs and it is proved that Griggs and Yeh’s conjecture is true for the homomorphic product of two graphs with minor exceptions.

Discrete Mathematics, 2004

A k-L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to {0,1,…,k} such that |f(u)−f(v)|⩾1 if d(u,v)=2 and |f(u)−f(v)|⩾2 if d(u,v)=1. The L(2,1)-labeling problem is to find the L(2,1)-labeling number λ(G) of a graph G which is the minimum cardinality k such that G has a k-L(2,1)-labeling. In this paper, we study L(2,1)-labeling numbers of

2016

A (2, 1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0, 1, . . . , k} of nonnegative integers such that | f (x)f (y)| ≥ 2 if x is a vertex and y is an edge incident to x, and | f (x)f (y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2, 1)-total labeling number λ T 2 (G) of a graph G is defined as the minimum k among all possible assignments. In [4], Chen and Wang conjectured that all outerplanar graphs G satisfy λ T 2 (G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G, while they also showed that it is true for G with ∆(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ T 2 (G) ≤ ∆(G) + 2 even in the case of ∆(G) ≤ 4.

Discussiones Mathematicae Graph Theory, 2007

Motivated by problems in radio channel assignments, we consider radio k-labelings of graphs. For a connected graph G and an integer k ≥ 1, a linear radio k-labeling of G is an assignment f of non negative integers to the vertices of G such that |f (x) − f (y)| ≥ k + 1 − d G (x, y), for any two distinct vertices x and y, where d G (x, y) is the distance between x and y in G. A cyclic k-labeling of G is defined analogously by using the cyclic metric on the labels. In both cases, we are interested in minimizing the span of the labeling. The linear (cyclic, respectively) radio k-labeling number of G is the minimum span of a linear (cyclic, respectively) radio k-labeling of G. In this paper, linear and cyclic radio k-labeling numbers of paths, stars and trees are studied. For the path P n of order n ≤ k + 1, we completely determine the cyclic and linear radio k-labeling numbers. For 1 ≤ k ≤ n − 2, a new improved lower bound for the linear radio k-labeling number is presented. Moreover, we give the exact value of the linear radio k-labeling number of stars and we present an upper bound for the linear radio k-labeling number of trees.

Discrete Mathematics, 2000

Given a graph G and a positive integer d, an L(d; 1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u) − f(v)|¿d; if u and v are not adjacent but there is a two-edge path between them, then

Journal of Applied Mathematics and Physics, 2023

An (,) L h k-labeling of a graph G is an assignment of non-negative integers to the vertices such that if two vertices u and v are adjacent then they receive labels that differ by at least h, and when u and v are not adjacent but there is a two-hop path between them, then they receive labels that differ by at least k. The span λ of such a labeling is the difference between the largest and the smallest vertex labels assigned. Let () k h G λ denote the least λ such that G admits an (,) L h k-labeling using labels from {0,1, , } λ . A Cayley graph of group is called circulant graph of order n, if the group is isomorphic to n . In this paper, initially we investigate the (,) L h k-labeling for circulant graphs with "large" connection sets, and then we extend our observation and find the span of (,) L h k-labeling for any circulants of order n.

Journal of Combinatorial Optimization, 2014

For positive numbers j and k, an L(j, k)-labeling f of G is an assignment of numbers to vertices of G such that |f (u) − f (v)| ≥ j if d(u, v) = 1, and |f (u) − f (v)| ≥ k if d(u, v) = 2. The span of f is the difference between the maximum and the minimum numbers assigned by f. The L(j, k)-labeling number of G, denoted by λ j,k (G), is the minimum span over all L(j, k)-labelings of G. In this article, we completely determine the L(j, k)-labeling number (2j ≤ k) of the Cartesian product of path and cycle.

Discrete Mathematics, Algorithms and Applications, 2021

An [Formula: see text]-labeling of a graph [Formula: see text] is an assignment of non-negative integers to the vertices such that if two vertices [Formula: see text] and [Formula: see text] are adjacent then they receive labels that differ by at least [Formula: see text], and when [Formula: see text] and [Formula: see text] are not adjacent but there is a two-hop path between them, then they receive labels that differ by at least one. The span [Formula: see text] of such a labeling is the difference between the largest and the smallest vertex labels assigned. Let [Formula: see text] denote the least [Formula: see text] such that [Formula: see text] admits an [Formula: see text]-labeling using labels from [Formula: see text]. A Cayley graph of group [Formula: see text] is called circulant graph of order [Formula: see text], if [Formula: see text]. In this paper initially we investigate the [Formula: see text]-labeling for circulant graphs with “large” connection sets, and then we ex...

Malaya Journal of Matematik

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers. In this paper, two new variations of labeling named k-distant edge total labeling and k-distant vertex total labeling are introduced. Moreover, the study of two new graph parameters, called k-distant edge chromatic number (γ kd) and k-distant vertex chromatic number (γ kd) related this labeling are initiated. The k-distant vertex total labeling for paths, cycles, complete graphs, stars, bi-stars and friendship graphs are studied and the value of the parameter γ kd determined for these graph classes. Then k-distant edge total labeling for paths, cycles and stars are studied. Also, an upper bound of γ kd and a lower bound of γ kd are presented for general graphs.

International Journal of Mobile Network Design and …, 2006

The (h, k)-coloring problem, better known as the L(h, k)-labelling problem, is that of vertex coloring an undirected graph with non-negative integers so that adjacent vertices receive colors that differ by at least h and vertices of distance 2 receive colors that differ by at least k. We give tight approximations for this problem on general graphs and on bipartite graphs