(2,1)-Total labelling of outerplanar graphs

The (2,1)(2,1)-total labelling number λ2T(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)Δ(G), then λ2T(G)⩽Δ(G)+2 if Δ(G)⩾5Δ(G)⩾5, or Δ(G)=3Δ(G)=3 and G   is 2-connected, or Δ(G)=4Δ(G)=4 and G contains no intersecting triangles.