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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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