(2,1)-Total labelling of outerplanar graphs

Abstract

The ( 2 , 1 ) -total labelling number λ 2 T ( G ) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least 2. In this paper we prove that if G is an outerplanar graph with maximum degree Δ ( G ) , then λ 2 T ( G ) ⩽ Δ ( G ) + 2 if Δ ( G ) ⩾ 5 , or Δ ( G ) = 3 and G is 2-connected, or Δ ( G ) = 4 and G contains no intersecting triangles.