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.