Some Results on Prime and Distinct Prime Distance Labeling of Graphs

Abstract

Let 𝒁 be the set of all integers. A graph 𝑯 is a prime distance graph if there exists an injective function 𝒈:𝑽(𝑯)→𝒁 such that for any two adjacent vertices 𝒙 and 𝒚, the integer |𝒈(𝒙)−𝒈(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 𝒈 is called 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≥𝟑) vertices with 𝑽(𝑷(𝒏,𝒌))={𝒗𝒊,𝒖𝒊:𝟎≤𝒊≤𝒏−𝟏}and 𝑬(𝑷(𝒏,𝒌))={𝒗𝒊𝒗𝒊+𝟏,𝒗𝒊𝒖𝒊,𝒖𝒊𝒖𝒊+𝒌:𝟎≤𝒊≤𝒏−𝟏, subscripts modulo 𝒏}. In this paper, we show 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≥𝟓 and all odd 𝒏≥𝟓, respectively. We also prove that the cycle 𝑪𝒏 admits a distinct prime distance labeling for all 𝒏≥𝟑, besides establishing the prime distance labeling for some graphs.