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Short disjoint paths in locally connected graphs. (English) Zbl 1127.05056
Summary: G. Chartrand and R. E. Pippert [Cas. Pest. Mat. 99, 158–163 (1974; Zbl 0278.05113)] proved that a connected, locally $$k$$-connected graph is $$(k + 1)$$-connected. We investigate the lengths of $$k + 1$$ disjoint paths between two vertices in locally $$k$$-connected graphs with respect to several graph parameters, e.g. the $$k$$-diameter of a graph. We also give a generalization of the mentioned result.
##### MSC:
 05C38 Paths and cycles 05C40 Connectivity
##### Keywords:
Connectivity; local connectivity; diameter; disjoint paths
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##### References:
 [1] Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, Macmillan, London, 1976 [2] Chartrand, G., Pippert, R.E.: Locally connected graphs, Časopis pro pěstování matematiky, 99, 158–163 (1974) · Zbl 0278.05113 [3] Frank, A.: Connectivity and network flows. In: Handbook of Combinatorics (R.L. Graham et al. eds.), Elsevier, Amsterdam, 1995, pp. 111–177 · Zbl 0846.05055 [4] Lovász, L., Neumann-Lara, V., Plummer, M.D.: Mengerian theorems for paths of bounded length, Period Math. Hungar. 9, 269–276 (1978) · Zbl 0393.05033 [5] Menger, K.: Zur allgemeinen Kurventheorie, Fund. Math. 10, 96–115 (1927) · JFM 53.0561.01
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