List version of ($p$,1)-total labellings

The (p,1)-total number λp (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 incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least p. In this paper we consider the list version. Let L(x) be a list of possible colors for all x ∈ V(G) ∪ E(G). Define Cp,1(G) to be the smallest integer k such that for every list assignment with |L(x)| = k for all x ∈ V(G) ∪ E(G), G has a (p,1)-total labelling c such that c(x) ∈ L(x) for all x ∈ V(G) ∪ E(G). We call Cp,1(G) the (p,1)-total labelling choosability and G is list L-(p,1)-total labelable. In this paper, we present a conjecture on the upper bound of Cp,1. Furthermore, we study this parameter for paths and trees in Section 2. We also prove that Cp,1(K1,n) ≤ n + 2p − 1 for star K1,n with p ≥ 2, n ≥ 3 in Section 3 and Cp,1(G) ≤ ∆ + 2p − 1 for outerplanar graph with ∆ ≥ p + 3 in Section 4. 2000 Mathematics Subject Classification : 05C15