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Connectivity of iterated line graphs. (English) Zbl 1009.05086
Summary: We present lower bounds for the connectivity of the \(i\)-iterated line graph \(L^i(G)\) of a graph \(G\). We prove that if \(G\) is a connected regular graph and \(i\geqslant 5\), then the connectivity of \(L^i(G)\) is equal to the degree of \(L^i(G)\), that is, the connectivity of \(L^i(G)\) attains its theoretical maximum (we remark that the bound on \(i\) is best possible). Moreover, if a hypothesis on the growth of the minimum degree of the \(i\)-iterated line graph is true, then an analogous result is true for an arbitrary graph \(G\) if \(i\) is sufficiently large.

MSC:
05C40 Connectivity
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