On (p, 1)-Total Labelling of Some 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. In this paper, it is proved that the (p, 1)-total labelling number (p ≥ 2) of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ (G) ≥ 6p + 7 or Δ (G) ≥ 4p + 6 and G is triangle-free.

Keywords

• 1-planar graph
• total coloring
• (p,1)-total labelling
• structural theorem
• 05c15
• 05c10