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Connectivity of iterated line graphs. (English) Zbl 1215.05097
Summary: Let \(k\geq 0\) be an integer and \(L^k(G)\) be the \(k\)th iterated line graph of a graph \(G\). Niepel and Knor proved that if \(G\) is a 4-connected graph, then \(\kappa (L^{2}(G))\geq 4\delta (G) - 6\). We show that the connectivity of \(G\) can be relaxed. In fact, we prove in this note that if \(G\) is an essentially 4-edge-connected and 3-connected graph, then \(\kappa (L^{2}(G))\geq 4\delta (G) - 6\). Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs.

MSC:
05C40 Connectivity
05C76 Graph operations (line graphs, products, etc.)
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References:
[1] Bondy, J.A.; Murty, U.S.R., Graph theory with applications, (1976), Macmillan London, Elsevier, New York · Zbl 1134.05001
[2] Chartrand, G.; Stewart, M.J., The connectivity of line graphs, Math. ann., 182, 170-174, (1969) · Zbl 0167.52203
[3] Knor, M.; Niepel, L’., Connectivity of iterated line graphs, Discrete appl. math., 125, 255-266, (2003) · Zbl 1009.05086
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