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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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