Radio D-distance number of some graphs (original) (raw)

On Radio Mean D-Distance Number of Graph Obtained from Graph Operation

International Journal of Mathematics Trends and Technology, 2018

A Radio Mean D-distance labeling of a connected graph G is an injective map f from the vertex set V(G) to ℕ such that for two distinct vertices u and v of G, d D (u, v) + + () 2 ≥ 1 + diam D (G), where d D (u, v) denotes the D-distance between u and v and diam D (G) denotes the D-diameter of G. The radio mean Ddistance number of f, rmn D (f) is the maximum label assigned to any vertex of G. The radio mean D-distance number of G, rmn D (G) is the minimum value of rmn D (f) taken over all radio mean D-distance labeling f of G. In this paper we find the radio mean D-distance number of graph obtained from graph operation.

Radio mean D-distance labeling of some graphs

A Radio Mean D-distance labeling of a connected graph G is an injective map f from the vertex set V(G) to ℕ such that for two distinct vertices u and v of G, d D (u, v) + í µí±“ í µí±¢ +í µí±“(í µí±£) 2 ≥ 1 + diam D (G), where d D (u, v) denotes the D-distance between u and v and diam D (G) denotes the D-diameter of G. The radio mean D-distance number of f, rmn D (f) is the maximum label assigned to any vertex of G. The radio mean D-distance number of G, rmn D (G) is the minimum value of rmn D (f) taken over all radio mean D-distance labeling f of G. In this paper we find the radio mean D-distance number of some well known graphs.

From rainbow to the lonely runner: a survey on coloring parameters of distance graphs

Taiwanese Journal of Mathematics, 2008

Motivated by the plane coloring problem, Eggleton, Erdős and Skelton initiated the study of distance graphs. Let D be a set of positive integers. The distance graph generated by D, denoted by G(Z, D), has all integers Z as the vertex set, and two vertices x and y are adjacent whenever |x − y| ∈ D. The chromatic number, circular chromatic number and fractional chromatic number of distance graphs have been studied extensively in the past two decades; these coloring parameters are also closely related to some problems studied in number theory and geometry. We survey some research advances and open problems on coloring parameters of distance graphs.

Distance connectivity in graphs and digraphs

1996

Let G = ( V , A ) be a digraph with diameter D # 1. For a given integer 2 5 t 5 D , the t-distance connectivity K ( t ) of G is the minimum cardinality of an z --+ y separating set over all the pairs of vertices z, y which are a t distance d(z, y) 2 t. The t-distance edge connectivity X ( t ) of G is defined similarly. The t-degree of G, h ( t ) , is the minimum among the out-degrees and in-degrees of all vertices with (out-or in-) eccentricity at least t. A digraph is said to be maximally distance connected if K ( t ) = 6 ( t ) for all values of t. In this paper we give a construction of a digraph having D -1 positive arbitrary integers c2 5 . . . 5 c D , D > 3, as the values of its t-distance connectivities 4 2 ) = cz, . . . , K ( D ) = cD.

On Distance Coloring

Given a graph G = (V, E), a (d, k)-coloring is a function from the vertices V to colors {1, 2, . . . , k} such that any two vertices within distance d of each other are assigned different colors. We determine the complexity of the (d, k)-coloring problem for all d and k, and enumerate some interesting properties of (d, k)-colorable graphs. Our main result is the discovery of a dichotomy between polynomial and NP-hard instances: for fixed d ≥ 2, the distance coloring problem is polynomial time for k ≤ 3d 2 and NP-hard for k > 3d 2 .

Studies in graph theory distance related concepts in graphs

2013

By a graph G = (V,E), we mean a finite undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n = |V | and m = |E| respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [7]. In Chapter 1, we collect some basic definitions and theorems on graphs which are needed for the subsequent chapters. The distance d(u, v) between two vertices u and v of a connected graph G is the length of a shortest u-v path in G. There are several distance related concepts and parameters such as eccentricity, radius, diameter, convexity and metric dimension which have been investigated by several authors in terms of theory and applications. An excellent treatment of various distances and distance related parameters are given in Buckley and Harary [6]. Let G = (V,E) be a graph. Let v ∈ V . The open neighborhood N(v) of a vertex v is the set of vertices adjacent to v. Thus N(v) = {w ∈ V : wv ∈ E}. The closed neighborhood of a vertex v, is the set...

On the d-distance face chromatic number of plane graphs

Discrete Mathematics, 1997

The d-distance face chromatic number of a connected plane graph G is the minimum number of colours in such a colouring of faces of G that whenever two distinct faces are at the distance at most d, they receive distinct colours. We estimate the d-distance face chromatic number from above for connected plane graphs with maximum degree at least eight.

On the distance connectivity of graphs and digraphs

Discrete Mathematics, 1994

Let G=( V, E) be a digraph with diameter D # 1. For a given integer 1 <t <II, the t-distance connectivity of G is the minimum cardinality of an x +y separating set over all the pairs of vertices x,y which are at distance d&y)> t. The t-distance edge-connectivity of G is defined analogously. This paper studies some results on the distance connectivities of digraphs and bipartite digraphs. These results are given in terms of the parameter I, which can be thought of as a generalization of the girth of a graph. For instance, it is proved that G is maximally connected iff either 0<21-1 or ~(21) > 6. As a corollary, similar results for (undirected) graphs are derived.

Vertex Coloring of Certain Distance Graphs

International Journal of Pure and Apllied Mathematics, 2013

In this paper first, we give a brief introduction about integer distance graphs. An integer distance graph is a graph G(Z, D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if |u − v| ∈ D where D is a subset of the positive integers. If D is a subset of P then we call G(Z, D) a prime distance graph. Second, we obtain a partial solution to a general open problem of characterizing a class of prime distance graphs. Third, we compute the vertex arboricity of certain prime distance graphs. Fourth, we give a brief review regarding circulant graphs and highlight its importance in the computation of chromatic number of distance graphs with appropriate references. Fifth, we introduce the notion of pseudochromatic coloring and obtain certain results concerning circulant graphs and distance graphs.