Inclusive Lucky Labeling of Graphs

we define a new notion called inclusive lucky labeling (ILL) and study proper inclusive lucky labeling (PILL) for simple undirected graphs. We also define inclusive lucky number and proper inclusive lucky number for some simple graphs. Keywords— Lucky labeling; proper lucky labeling; inclusive lucky labeling; proper inclusive lucky labeling; Inclusive lucky number; proper inclusive lucky number


I. INTRODUCTION
A graph labeling is an assignment of labels (or integers) to vertices or edges or both. A label is said to be proper if no two adjacent vertices have same label. The concept of labeling was introduced by Rosa [4] in 1967. It was further developed by Graham and Sloane in 1980 [3].
Several types of labeling were introduced and studied for different types of graphs. The concept of Lucky labeling of graphs were studied by A. Ahadi et al [1] and S. Akbari et al [2]. Lucky number for different kinds of graphs has been studied.
In this paper, we define a new notion of labeling called inclusive lucky labeling (ILL). We also define proper inclusive lucky labeling (PILL), inclusive lucky number and proper inclusive lucky number for some simple undirected graphs.

II. PRELIMINARIES
In this section, we provide some basic definitions and results as in [1,2]. Let G be a simple graph with vertex set V(G). For any vertex v in G, N(v) denotes the set of all the vertices which are adjacent with v and N[v]=N(v)U{v}. Definition 1. Let f:V(G)→{1, 2, 3, …} be a labeling of the vertices of a graph G and let S(v) denote the sum of labels over the neighbors of the vertex v in G. For an isolated vertex v, S(v) = 0.
A labeling f is said to be a lucky labeling or simply lucky if S(u)≠S(v) for every pair of adjacent vertices u and v. The least positive integer k for which there exists a lucky labeling f: V(G)→{1, 2, 3, …, k} for the graph G is called lucky number of a graph G and it is denoted by η(G). Definition 2. A labeling f is said to be proper lucky labeling if f is lucky and proper. The proper lucky number ηp(G) of a graph G is the least positive integer k for which G has a proper lucky labeling with {1, 2, 3, .., k} as the set of labels. Example: For complete graph Kn, η(Kn)=2 and ηp(Kn)=4. Result: For any connected graph G, η(G)≤ ηp(G).

III. INCLUSIVE LUCKY LABELING
In this section, we define a new notion of labeling called inclusive lucky labeling and proper inclusive lucky labeling. We also define the inclusive lucky number and proper inclusive lucky number. Further, we discuss the inclusive lucky number and proper inclusive number of some standard graphs. Definition

, … , k} is called proper inclusive lucky number of a graph G and it is denoted by ηip(G).
A graph which admits an inclusive lucky labeling is called ILL graph and a graph which admits a proper inclusive lucky labeling is called PILL graph.
Theorem 1: For n ≥ 2, the complete graph Kn is not ILL graph and it not PILL graph. Proof: In a complete graph any two vertices are adjacent and so for any vertex v, Thus there is no inclusive lucky labeling and proper inclusive lucking labeling for Kn and hence Kn is not an ILL graph and also not a PILL graph. Remark: The complete graphs are lucky but not inclusive lucky. This shows that the lucky labeling and inclusive lucky labeling are different. . Therefore, f : V (G)→{1, 2} is an inclusive lucky labeling of G and so P3 is a ILL graph and ηi(P3) = 1.
The inclusive labeling defined above is not proper and so ηip(P3) ≥ 2. Label the internal vertex as 2 and the end vertices with 1, we get a proper inclusive lucky labeling 2 labels. Therefore, P3 is a PILL graph and ηip(P3) = 2. Definition: (Star graph Sn) A star graph Sn is a graph obtained by joining n pendent edges to a single vertex.
It has n + 1 vertices and n edges. For n ≥ 2, C2n+1 has atleast two internal vertices and so an inclusive lucky labeling with {1} is not possible. Any labeling of C2n+1 with {1, 2} contains two adjacent vertices u and v such that S[u]=S [v]. Therefore, such labeling is not an inclusive lucky labeling and so ηi(C2n+1) ≥ 3.