Prime Distance Graphs and Applications of Graph Labeling: A Modelling Approach

Abstract

A graph H is a prime distance graph if there exists an one-to-one function h : V (H ) \(\to\) Z such that for any two adjacent vertices x and y, the integer |h(x) - h(y)| is a prime. So H is a prime distance graph if and only if there exists a prime distance labeling of H. If the edge labels of H are also distinct, then h becomes a distinct prime distance labeling of H and H is called a distinct prime distance graph. The generalized Petersen graphs P (n, k) are defined to be a graph on 2n (n \(\ge\) 3) vertices with V (P (n, k)) = {vi , ui : 0 \(\le\) i \(\le\)n - 1} and E(P (n, k)) = {vi vi+1 , vi ui , ui ui+k : 0 \(\le\) i \(\le\) n - 1, subscripts modulo n}. In this chapter, we give a concise overview on these two labeling, specifically, highlight that the generalized Petersen graphs P (n, 3) permit a prime distance labeling for all even n > 5 and conjecture that P (n, 2) and P (n, 3) admit a prime distance labeling for any n \(\ge\) 5 and all odd n \(\ge\) 5, respectively. Moreover, the cycle Cn admits a distinct prime distance labeling for all n \(\ge\) 3.