On \(\text{Ł}(\wp, \wp - 1, \dots, 1)\) labelings of circulant graphs
Summary: An \(m\)-\(\text{Ł}(\wp, \wp - 1, \dots, 1)\) labeling of a simple graph \(\mathscr{G}\) is a mapping \(\tau\) from the vertex set \(\vee(\mathscr{G})\) to \(\{0, 1, \dots, m\}\) such that \(|\tau(u) - \tau(v)|\geq\wp+1-\mathfrak{d}(u, v)\), \(\forall u,v\in\vee(\mathscr{G})\), where length of the shortest route connecting \(u\) and \(v\) is represented by \(\mathfrak{d}(u, v)\). The smallest \(m\) for which there exists a \(m\)-\(\text{Ł}(\wp, \wp - 1, \dots, 1)\) labeling of \(\mathscr{G}\) is known as the \(\text{Ł}(\wp,\wp - 1, \dots, 1)\) labeling number of \(\mathscr{G}\), and it is described by \(\lambda_\wp(\mathscr{G})\). We define \(m\)-\(\text{Ł}^\prime(\wp,\wp - 1, \dots, 1)\) as the same as the \(m\)-\(\text{Ł}(\wp, \wp - 1, \dots, 1)\) labeling if \(\tau\) is one to one. The \(\text{Ł}^\prime(\wp, \wp - 1, \dots, 1)\) labeling number of \(\mathscr{G}\) represented by \(\lambda(\wp, \wp - 1, \dots, 1)_\prime( \mathscr{G})\) and is called minimum span. In this paper, we prove that the circulant graphs with specified generating sets admit \(m\)-\(\text{Ł}(\wp, \wp - 1, \dots, 1)\) and \(m\)-\(\text{Ł}^\prime(\wp, \wp - 1, \dots, 1)\) labeling and also find \(\lambda_\wp(\mathscr{G})\) and \(\lambda (\wp ,\wp - 1, \dots, 1)^\prime(\mathscr{G})\). Moreover, we find the \(\text{Ł}(\wp, \wp - 1, \dots, 1)\) labeling number of any simple graph with diameter less than \(\wp\).
scientific article; zbMATH DE number 7433088