(p, 1)-Total labelling of graphs

A (p,1)-total labelling of a graph G is an assignment of integers to V(G)@?E(G) such that: (i) any two adjacent vertices of G receive distinct integers, (ii) any two adjacent edges of G receive distinct integers, and (iii) a vertex and its incident edge receive integers that differ by at least p in absolute value. The span of a (p,1)-total labelling is the maximum difference between two labels. The minimum span of a (p,1)-total labelling of G is called the (p,1)-total number and denoted by @l"p^T(G). We provide lower and upper bounds for the (p,1)-total number. In particular, generalizing the Total Colouring Conjecture, we conjecture that @l"p^T=<@D+2p-1 and give some evidences to support it. Finally, we determine the exact value of @l"p^T(K"n), except for even n in the interval [p+5,6p^2-10p+4] for which we show that @l"p^T(K"n)@?{n+2p-3,n+2p-2}.