The (p,1)-total number Ξ»pTβ(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 xβV(G)βͺE(G). Define
Cp,1Tβ(G) to be the smallest integer k such that for every list
assignment with β£L(x)β£=k for all xβV(G)βͺE(G), G has a
(p,1)-total labelling c such that c(x)βL(x) for all xβV(G)βͺE(G). We call Cp,1Tβ(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,1Tβ. Furthermore, we study this parameter for paths
and trees in Section 2. We also prove that Cp,1Tβ(K1,nβ)β€n+2pβ1 for
star K1,nβ with pβ₯2,nβ₯3 in Section 3 and Cp,1Tβ(G)β€Ξ+2pβ1 for outerplanar graph with Ξβ₯p+3 in Section 4.Comment: 11 pages, 2 figure