Prime labeling of special graphs

M D M C P Weerarathna,T R D S M Thennakoon,A A I Perera,K D E Dhananjaya

doi:10.4038/jsc.v12i1.30

Abstract

Prime labeling is the most interesting category of graph labeling with various applications. A graph G= (V(G), E(G)) with |v(G)| vertices are said to have prime labeling if its vertices are labeled with distinct positive integers 1,2,3,……,|v| such that for each edge uv e E(G) the labels assigned to u and v are relatively prime, where V(G) and E(G) are vertex set and edge set of G, respectively. Therefore, the graph G has a prime labeling whenever any of two adjacent vertices can be labeled as two relative prime numbers and is called a prime graph. In our work, we focus on the prime labeling method for newly constructed graphs obtained by replacing each edge of a star graph K1,n by a complete tripartite graph K1,m,1 for m = 2,3,4, and 5, which are prime graphs. In addition to that, investigate another type of simple undirected finite graphs generalized by using circular ladder graphs. These new graphs obtained by attaching K1,2 at each external vertex of the circular ladder graph CLn and proved that the constructed graphs are prime graphs when n ≥ 3 and n ≠ 1 (mod3) . Finally, focus on another particular type of simple undirected finite graph called a scorpion graph, denoted by S(2p,2q,r) . The Scorpion graph gets its name from shape, which resembles a scorpion, having 2p + 2q + r vertices p ≥ 1, q ≥ 2, r ≥ 2)are placed in the head, body, and tail respectively. To prove that the scorpion graph has prime labeling, we used two results that have already been proved for ladder graphs.

Highlights

Graph labeling is a prominent research area in Graph theory, and there are considerable number of open problems and literature are available for various types of graphs
In Theorem and Theorem, we discuss the prime labeling method of graph obtained by replacing every edge of a star graph by the tripartite graph
Prime labeling is the most valuable area of graph labeling with various applications

Summary

Introduction

Graph labeling is a prominent research area in Graph theory, and there are considerable number of open problems and literature are available for various types of graphs. Distinct positive integers are assigned to the vertices which are less than or equal to the number of vertices in the graph such that labels of adjacent vertices are relatively prime. Edge labeling is another particular area in graph labeling and was introduced by Deretsky, Lee, and Mitchem in 1991. Throughout this paper, we considered the vertex prime labeling for a new type of graphs. We consider prime labeling of newly constructed graphs obtained by replacing every edge of a star graph by the tripartite graph for and , Stripe Blade Fan Graph

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