Taiwanese Journal of Mathematics

The IC-indices of Complete Multipartite Graphs

Chin-Lin Shiue, Hui-Chuan Lu, and Jun-yi Kuo

Full-text: Open access

Abstract

Given a connected graph $G$, a function $f$ mapping the vertex set of $G$ into the set of all integers is a coloring of $G$. For any subgraph $H$ of $G$, we denote as $f(H)$ the sum of the values of $f$ on the vertices of $H$. If for any integer $k \in \{1,2,\ldots,f(G)\}$, there exists an induced connected subgraph $H$ of $G$ such that $f(H) = k$, then the coloring $f$ is called an IC-coloring of $G$. The IC-index of $G$, written $M(G)$, is defined to be the maximum value of $f(G)$ over all possible IC-colorings $f$ of $G$. In this paper, we give a lower bound on the IC-index of any complete $\ell$-partite graph for all $\ell \geq 3$ and then show that, when $\ell = 3$, our lower bound also serves as an upper bound. As a consequence, the exact value of the IC-index of any tripartite graph is determined.

Article information

Source
Taiwanese J. Math., Volume 21, Number 6 (2017), 1213-1231.

Dates
Received: 5 October 2016
Revised: 14 March 2017
Accepted: 16 March 2017
First available in Project Euclid: 17 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1502935247

Digital Object Identifier
doi:10.11650/tjm/8031

Mathematical Reviews number (MathSciNet)
MR3732903

Zentralblatt MATH identifier
06871366

Subjects
Primary: 05C15: Coloring of graphs and hypergraphs

Keywords
IC-coloring IC-index complete multipartite graph complete tripartite graph

Citation

Shiue, Chin-Lin; Lu, Hui-Chuan; Kuo, Jun-yi. The IC-indices of Complete Multipartite Graphs. Taiwanese J. Math. 21 (2017), no. 6, 1213--1231. doi:10.11650/tjm/8031. https://projecteuclid.org/euclid.twjm/1502935247


Export citation

References

  • R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly 87 (1980), no. 3, 206–210. https://doi.org/10.2307/2321610
  • R. K. Guy, Unsolved Problems in Number Theory, Second edition, Problem Books in Mathematics, Unsolved Problems in Intuitive Mathematics I, Springer-Verlag, New York, 1994. https://doi.org/10.1007/978-1-4899-3585-4
  • R. L. Heimer and H. Langenbach, The stamp problem, J. Recreational Math. 7 (1974), no. 3, 235–250.
  • C. T. Long and N. Woo, On bases for the set of integers, Duke Math. J. 38 (1971), 583–590. https://doi.org/10.1215/s0012-7094-71-03872-5
  • W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969), no. 4, 377–380. https://doi.org/10.1093/comjnl/12.4.377
  • S. Mossige, The postage stamp problem: an algorithm to determine the $h$-range on the $h$-range formula on the extremal basis problem for $k = 4$, Math. Comp. 69 (2000), no. 229, 325–337. https://doi.org/10.1090/s0025-5718-99-01204-1
  • S. G. Penrice, Some new graph labeling problems: a preliminary report, DIMACS Tech. Rep. 95-26 (1995), 1–9.
  • E. Salehi, S.-M. Lee and M. Khatirinejad, IC-colorings and IC-indices of graphs, Discrete Math. 299 (2005), no. 1-3, 297–310. https://doi.org/10.1016/j.disc.2004.02.026
  • E. S. Selmer, On the postage stamp problem with the three stamp denominations, Math. Scand. 47 (1980), no. 1, 29–71. https://doi.org/10.7146/math.scand.a-11874
  • C.-L. Shiue and H.-L. Fu, The IC-indices of complete bipartite graphs, Electron. J. Combin. 15 (2008), no. 1, Research paper 43, 13 pp.
  • C.-L. Shiue and H.-C Lu, The IC-indices of some complete multipartite graphs. arXiv:1610.00238
  • R. G. Stanton, J. A. Bate and R. C. Mullin, Some tables for the postage stamp problem, Proceedings of the Fourth Manitoba Conference on Numerical Mathematics (Winnipeg, Man., 1974), 351–356, Utilitas Math., Winnipeg, Man., 1975.
  • A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe I, II, J. Reine Angew. Math. 194 (1955), 40–65, 111–140. https://doi.org/10.1515/crll.1955.194.40 https://doi.org/10.1515/crll.1955.194.111
  • D. B. West, Introduction to Graph Theory, Prentice-Hall, Upper Saddle River, NJ, 1996.