Abstract
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 \(4\lfloor\frac{p}{2}\rfloor+3\leq n\leq4p-1\). 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.
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Acknowledgements
The authors thank the referees for helpful comments which resulted in an improvement in the clarity of exposition of the paper. All authors were supported in part by the National Science Council under grants NSC97-2115-M-156-004-MY2 (M.-L.C.), NSC97-2115-M-259-002-MY3 (D.K.), NSC94-2115-M-156-001 (J.-H.Y.).
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Chia, ML., Kuo, D., Yan, JH. et al. (p,q)-total labeling of complete graphs. J Comb Optim 25, 543–561 (2013). https://doi.org/10.1007/s10878-012-9471-1
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DOI: https://doi.org/10.1007/s10878-012-9471-1