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December, 2017 The IC-indices of Complete Multipartite Graphs
Chin-Lin Shiue, Hui-Chuan Lu, Jun-yi Kuo
Taiwanese J. Math. 21(6): 1213-1231 (December, 2017). DOI: 10.11650/tjm/8031


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.


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Chin-Lin Shiue. Hui-Chuan Lu. Jun-yi Kuo. "The IC-indices of Complete Multipartite Graphs." Taiwanese J. Math. 21 (6) 1213 - 1231, December, 2017.


Received: 5 October 2016; Revised: 14 March 2017; Accepted: 16 March 2017; Published: December, 2017
First available in Project Euclid: 17 August 2017

zbMATH: 06871366
MathSciNet: MR3732903
Digital Object Identifier: 10.11650/tjm/8031

Primary: 05C15

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

Rights: Copyright © 2017 The Mathematical Society of the Republic of China


Vol.21 • No. 6 • December, 2017
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