On the k-friendly index of graphs

Discrete Mathematics, 2011

Let G be a graph with vertex set V(G) and edge set E(G), and f be a 0−1 labeling of E(G) so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling fedge-friendly. We say an edge-friendly labeling induces a partial vertex labeling if vertices which are incident

Discrete Mathematics, 1994

This paper deals with certain edge labelings of graphs. After having introduced the concepts of a weak antimagic graph and an Egyptian magic graph, the authors showed that every connected graph of order 23 is weakly antimagic and proved some interesting relations between the different edge labelings. * Corresponding author 0012-365X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved SSDl 0012-365X(93)E0054-8

For a positive integer $n$, let $\mZ$ be the set of all non-negative integers modulo $n$ and $\sP(\mZ)$ be its power set. A sumset valuation or a sumset labeling of a given graph $G$ is an injective function $f:V(G) \to \sP(\mZ)$ such that the induced function $f^+:E(G) \to \sP(\mZ)$ defined by $f^+ (uv) = f(u)+ f(v)$. A sumset indexer of a graph $G$ is an injective sumset valued function $f:V(G) \to \sP(\mZ)$ such that the induced function $f^+:E(G) \to \sP(\mZ)$ is also injective. In this paper, some properties and characteristics of this type of sumset labeling of graphs are being studied.

rose-hulman.edu

Interference between radio signals can be modeled using distance labeling where the vertices on the graph represent the radio towers and the edges represent the interference between the towers. The distance between vertices affects the labeling of the vertices to account for the strength of interference. In this paper we consider three levels of interference between signals on a given graph, G. Define D(x, y) to represent the distance between vertex x and vertex y. An L(d, j, s) labeling of graph G is a function f from the vertex set of a graph to the set of positive integers, where |f (

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.

2008

Let G be a graph with vertex set V(G) and edge set E(G), and let A ={O, I}. A labeling f: V(G) -7 A induces a partial edge labeling f* : E(G) -7 A defined by f*(xy) = f{x), if and only iff{x) f{y), for each edge xy E E(G). For i E A, let vt<i) = card{v E V(G): f{v) i} and er-(i) card{e E E(G): f"(e) = i}. A labeling f of a graph G is said to be friendly if I vt<0) vt< I) I=: ;; 1. If, let< 0) et<l) I =:;; I then G is said to be balanced. The balance index set of the graph G, Bl(G), is defined as {let<O) -et<1)1 : the vertex labeling fis friendly}. Results parallel to the concept offriendly index sets are presented.

Anais do VI Encontro de Teoria da Computação (ETC 2021), 2021

We compare the behaviour of the $L(h,k)$-number of undirected and oriented graphs in terms of maximum degree, highlighting differences between the two contexts. In particular, we prove that, for every $h$ and $k$, oriented graphs with bounded degree in every block of their underlying graph (for instance, oriented trees and oriented cacti) have bounded $L(h,k)$-number, giving an upper bound on this number which is sharp up to a multiplicative factor $4$.

DOAJ (DOAJ: Directory of Open Access Journals), 2012

Let G = (V, E) be a connected simple graph. A labeling f : V → Z2 induces two edge labelings f + , f * : E → Z2 defined by f + (xy) = f (x)+f (y) and f * (xy) = f (x)f (y) for each xy ∈ E. For i ∈ Z2, let v f (i) = |f −1 (i)|, e f + (i) = |(f +) −1 (i)| and e f * (i) = |(f *) −1 (i)|. A labeling f is called friendly if |v f (1) − v f (0)| ≤ 1. For a friendly labeling f of a graph G, the friendly index of G under f is defined by i + f (G) = e f + (1) − e f + (0). The set {i + f (G) | f is a friendly labeling ofG} is called the full friendly index set of G. Also, the product-cordial index of G under f is defined by i * f (G) = e f * (1) − e f * (0). The set {i * f (G) | f is a friendly labeling ofG} is called the full product-cordial index set of G. In this paper, we find a relation between the friendly index and the product-cordial index of a regular graph. As applications, we will determine the full product-cordial index sets of torus graphs which was asked by Kwong, Lee and Ng in 2010; and those of cycles.

Mathematics and Statistics, 2020

Graph labeling is an assignment of integers to the vertices or the edges, or both, subject to certain conditions. In literature we find several labelings such as graceful, harmonious, binary, friendly, cordial, ternary and many more. A friendly labeling is a binary mapping such that |v f (1) -v f (0)|≤ 1 where v f (1) and v f (0) represents number of vertices labeled by 1 and 0 respectively. For each edge uv where e f * (1) and e f * (0) are the number of edges labeled 1 and 0 respectively. A friendly index set of a graph is {|e f * (1)e f * (0)|: f * runs over all f riendly labeling f of G} and it is denoted by F I(G). A mapping f : V (G) → {0, 1, 2} is called ternary vertex labeling and f (v) represents the vertex label for v. In this article, we extend the concept of ternary vertex labeling to 3-vertex friendly labeling and define 3-vertex friendly index set of graphs. The set F I 3v (G) = {|e f * (i) -e f * (j)|: f * runs over all 3vertex f riendly labeling f f or all i, j ∈ {0, 1, 2}} is referred as 3-vertex friendly index set. In order to achieve F I 3v (G), number of vertices are partitioned into {V 0 , V 1 , V 2 } such that ||V i |-|V j ||≤ 1 for all i, j = 0, 1, 2 with i = j and label the edge uv by |f In this paper, we study the 3-vertex friendly index sets of some standard graphs such as complete graph K n , path P n , wheel graph W n , complete bipartite graph K m,n and cycle with parallel chords P C n .

Advances in Mathematics: Scientific Journal, 2020

After a long period of scramble over analysis and investigations, the notion of graceful labeling came into existence. A mapping f for a graph G = (R, S) is said to be graceful if there exists a bijective mapping f : R (G) → N ∪ {0} such that each edge has an induced label ω (f, R (G)) = {|f (u) − f (v)| : u, v R (G)} and the resulting edge labels are distinct. In this paper, we introduce a new type of graph labeling for a graph G = (R, S) which we call G-graceful labeling. The G-graceful labeling for the graph G = (R, S) with r vertices and sedges is an injective function ρ : R(G) → {0, 1, 2, 3,. .. , t − 1}such that the induced function ρ * : S(G) → N is given byρ * (r, s) = {ρ * (r) + ρ * (s)},the resulting edge label are distinct. In this paper, we also prove that the graphs: path, ladder graph, flower graph, complete bipartite graph and star graph admit G-graceful labeling.

Jordan Journal of Mathematics and Statistics, 2024

In this paper, we introduce two new concepts namely bijective product k-cordial labeling and bijective product square kcordial labeling and show that some standard graphs admit bijective product k-cordial labeling, where k = 2, 3. Also, we establish that the path, cycle, flower, helm and gear graphs are bijective product square 3-cordial graphs.

Let Z2 = {0, 1} and G = (V, E) be a graph. A labeling f : V −→ Z2 induces an edge labeling f * : E −→ Z2 defined by f * (uv) = f (u).f (v). For i ∈ Z2, let v f (i) = v(i) = card{v ∈ V : f (v) = i} and e f (i) = e(i) = card{e ∈ E : f * (e) = i}. A labeling f is said to be vertex-friendly if | v(0) − v(1) |≤ 1. The vertex balance index set is defined by {| e f (0) − e f (1) | : f is vertex-friendly}. In this paper we completely determine the vertex balance index set of Kn, Km,n, Cn × P2 and Complete binary tree.

Discrete Mathematics, 2008

Let G=(V , E) be a graph, a vertex labeling f : V → Z 2 induces an edge labeling f * : E → Z 2 defined by f * (xy)=f (x)+f (y) for each xy ∈ E. For each i ∈ Z 2 , define v f (i) = |f −1 (i)| and e f (i) = |f * −1 (i)|. We call f friendly if |v f (1) − v f (0)| 1. The full friendly index set of G is the set of all possible values of e f (1) − e f (0), where f is friendly. In this note, we study the full friendly index set of the grid graph P 2 × P n .

2014

In the mathematical discipline of Graph Theory, labeling of graphs is a mapping that sends the edges or vertices or both of a graph to the set of numbers or subset of a set under certain conditions. Ever since the introduction of the concept labeling of graphs, it has been an active area of research in Graph Theory. Most graph labelings trace their origin to a paper by Rosa [14]. In 1983, B. D. Acharya [1] initiated a general study of the labeling of the vertices and edges of a graph using the subset of a set known as set-valuation of graphs and indicated their potential applications in different areas of human inquiry. He introduced the concept of set-indexer of a graph and proved that every graph admits a set-indexer. Acharya also pioneered the notion of set-indexing number of a graph developing the classes of set-graceful as well as set-semigraceful graphs. Meanwhile, Acharya and Hegde [5] undertook the study of yet another notion of set-valuation of graphs called set-sequential ...

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