(p,q)-total labeling of complete graphs

Ma-Lian Chia (), David Kuo (), Jing-Ho Yan () and Siou-Ru Yang ()
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Ma-Lian Chia: Aletheia University
David Kuo: National Dong Hwa University
Jing-Ho Yan: Aletheia University

Journal of Combinatorial Optimization, 2013, vol. 25, issue 4, No 6, 543-561

Abstract: 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

Keywords: L(2; 1)-labeling; (p; q)-total labeling; Complete graph; Matching (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-012-9471-1

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