Labeling Circulant Graphs: Distance Two Condition

Abstract

Given n≥6, D={1,2,…,⌊n2⌋}, and a generating set S⊆D, the circulant graph Cn(S) has Zn as a vertex set in which two distinct vertices i and j are adjacent if and only if |i−j|n∈S, where |x|n=min(|x|,n−|x|) is the circular distance modulo n. In this paper, we determine the L(2,1)-labeling number of Cn(D∖X), referred to as λ(Cn(D∖X)), for X={⌊n2⌋}, X={a}, X={a,b}, and in the general case when |X|<⌊n2⌋−⌈n4⌉, where a,b∈D. Furthermore, we demonstrate that for all n≥6 and any given set S, λ(Cn(S))=n+gcd(n,S¯)−2 if and only if gcd(n,S¯)≥2, and λ(Cn(S))≤n−1 if and only if gcd(n,S¯)=1. Additionally, we establish that when the diameter of Cn(S) equals 2, λ(Cn(S))=n−1. This observation motivated us to investigate the properties of S that lead to a diameter of Cn(S) equal to 2. Then, we introduce a highly distinctive family, denoted as An, that generates a large number of generating sets. For each value of n, we acquire a circulant graph Cn(An) with a diameter of 2, λ(Cn(An))=n−1, and various additional interesting properties.