## Taiwanese Journal of Mathematics

### The IC-indices of Complete Multipartite Graphs

#### 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
Revised: 14 March 2017
Accepted: 16 March 2017
First available in Project Euclid: 17 August 2017

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

#### 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

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