(p,q)-total labeling of complete graphs

Given a graph G and positive integers p,q with p≥q, the (p,q)-total number $lambda_{p,q}^{T}(G)$ of G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that the labels of any two adjacent vertices are at least q apart, the labels of any two adjacent edges are at least q apart, and the difference between the labels of a vertex and its incident edges is at least p. Havet and Yu (Discrete Math 308:496–513, 2008) first introduced this problem and determined the exact value of $lambda_{p,1}^{T}(K_{n})$ except for even n with p+5≤n≤6p ²?10p+4. Their proof for showing that $lambda _{p,1}^{T}(K_{n})leq n+2p-3$ for odd n has some mistakes. In this paper, we prove that if n is odd, then $lambda_{p}^{T}(K_{n})leq n+2p-3$ if p=2, p=3, or $4lfloorfrac{p}{2}rfloor+3leq nleq4p-1$ . And we extend some results that were given in Havet and Yu (Discrete Math 308:496–513, 2008). Beside these, we give a lower bound for $lambda_{p,q}^{T}(K_{n})$ under the condition that q<p<2q.