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Multi-level and antipodal labelings for certain classes of circulant graphs. (English) Zbl 1338.05066

Summary: A radio \(k\)-labeling \(c\) of a graph \(G\) is a mapping \(c : V(G)\to Z^+ \cup \{0\}\) such that \(d(u,v)+|c(u)-c(v)| \geq k+1\) for every two distinct vertices \(u\) and \(v\) of \(G\), where \(d(u,v)\) is the distance between any two vertices \(u\) and \(v\) of \(G\). The span of a radio \(k\)-labeling \(c\) is denoted by \(sp(c)\) and defined as \(\max \{|c(u)-c(v)| : u, v\in V(G)\}\). The radio labeling is a radio \(k\)-labeling when \(k = \mathrm{diam}(G)\). In other words, a radio labeling is a one-to-one function \(f\) from \(V(G)\) to \(Z^+\cup \{0\}\) such that \(|c(u)-c(v)|\geq \mathrm{diam}(G)+1-d(u,v)\) for any pair of vertices \(u\), \(v\) in \(G\). The radio number of \(G\) expressed by \(\mathrm{rn}(G)\), is the lowest span taken over all radio labelings of the graph. For \(k = \mathrm{diam}(G)-1\), a radio \(k\)-labeling is called a radio antipodal labeling. An antipodal labeling for a graph \(G\) is a function \(c : V(G)\to \{0,1,2,\ldots\}\) such that \(d(u,v)+|c(u)-c(v)|\geq \mathrm{diam}(G)\) for all \(u, v\in V(G)\). The radio antipodal number for \(G\) denoted by \(\mathrm{an}(G)\), is the minimum span of an antipodal labeling admitted by \(G\). In this paper, we investigate the exact value of the radio number and radio antipodal number for the circulant graphs \(G(4mk+2m;\{1,2m\})\), when \(m\geq 3\) is odd. Furthermore, we also determine the lower bound of the radio number for the circulant graphs \(G(4mk+2m;\{1,2m\})\), when \(m\geq 2\) is even.

MSC:

05C12 Distance in graphs
05C15 Coloring of graphs and hypergraphs
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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