Labeling of chordal rings (Q2848788)

A chordal ring \(CR_n(x,y,z)\) is the graph \(G = (V,E)\) where \(x,y,z \in \{1,2,\dots,n-1\}\) are distinct odd numbers, \(V= {\mathbb{Z}}_n\) and \(E = \{(i,i+x \mod n), (i,i+y \mod n),(i,i+z \mod n)|i = 2k,\;k\in\{0,1,2,\dots,\lfloor{(n-1)/2}\rfloor\}\}\). A super vertex-magic total labeling of a graph \(G=(V,E)\) is a one-to-one map \(\lambda : V \cup E \rightarrow \{1,2,\dots,|V|+|E|\}\) such that \(\lambda(V) = \{1,,2,\dots,|V|\}\), \(\lambda(E) = \{|V|+1,|V|+2,\dots,|V|+|E|\}\) and \(\lambda(x)+\sum_{xy\in E}{\lambda(xy)} = C\) for some constant \(C\), called a magic constant for \(\lambda\). An \((a,d)\)-antimagic labeling of \(G\) is a one-to-one map \(\lambda : E \rightarrow \{1,2,\dots,|E|\}\) such that the induced vertex labeling \(g_\lambda\), \(g_\lambda(x) = \sum_{xy\in E}{\lambda(xy)}\) is injective and \(g_\lambda(V) = \{a+kd | k=0,1,2,\dots,|V|-1\}\), where \(a>0\) and \(d \geq 0\) are fixed integers. In the paper, a super vertex-magic magic labeling of \(CR_n(1,3,n-1) \cong CR_n(1,3,5)\), \(n \equiv 0 \pmod 4\) is provided. The authors also present an \((a,d)\)-antimagic labeling of \(CR_n(1,5,n-1) \cong CR_n(1,3,7)\), \(n \equiv 0 \pmod 4\).