On \((p,1)\)-Total Labelling of Special \(1\)-Planar Graphs

Abstract

A graph is \(1\)-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. A \(k\)-(p, 1)-total labelling of a graph \(G\) is a function \(f\) from \(V(G) \cup E(G)\) to the color set \(\{0, 1, \ldots, k\}\) such that \(|f(u) – f(v)| \geq 1\) if \(uv \in E(G)\), \(|f(e_1) – f(e_2)| \geq 1\) if \(e_1\) and \(e_2\) are two adjacent edges in \(G\), and \(|f(u) – f(e)| \geq p\) if the vertex \(u\) is incident to the edge \(e\). The minimum \(k\) such that \(G\) has a \(k-(p, 1)\)-total labelling, denoted by \(\lambda_p^T(G)\), is called the \((p, 1)\)-total labelling number of \(G\). In this paper, we prove that, if a 1-planar graph \(G\) satisfies that maximum degree \(\Delta(G) \geq 7p + 1\) and no adjacent triangles in \(G\) or maximum degree \(\Delta(G) \geq 6p + 3\) and no intersecting triangles in \(G\), then \(\lambda_p^T(G) \leq \Delta + 2p – 2\), \(p \geq 2\).