## The IC-indices of complete multipartite graphs.(English)Zbl 1387.05097

Summary: 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.

### MSC:

 05C15 Coloring of graphs and hypergraphs 05C40 Connectivity
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### References:

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