Affiliation:
1. Polydisciplinary Faculty of Safi, Cadi Ayyad University, B.P. 4162, Sidi Bouzid, Safi 46000, Morocco
2. Faculty of Sciences and Technologies of Settat, Hassan First, University of Settat, Road to Casablanca, Km 3,5 B.P. 577, Settat 26000, Morocco
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.
Subject
Physics and Astronomy (miscellaneous),General Mathematics,Chemistry (miscellaneous),Computer Science (miscellaneous)
Reference26 articles.
1. Radio antipodal colorings of graphs;Chartrand;Math. Bohem.,2002
2. Radio k-colorings of paths;Chartrand;Discuss. Math. Graph Theory,2004
3. Radio k-labelings for cartesian products of graphs;Kchikech;Discuss. Math. Graph Theory,2008
4. Measures of traceability in graphs;Saenpholphat;Math. Bohem.,2006
5. Upper Hamiltonian numbers and Hamiltonian spectra of graphs;Tong;Australas. J. Comb.,2006