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On the $$\ell$$-connectivity of a graph. (English) Zbl 0631.05031
Let be $$\ell \geq 2$$, then the $$\ell$$-connectivity of a graph G, $$\kappa_{\ell}(G)$$, in the minimum number of vertices whose removal produces a disconnected graph with at least $$\ell$$ components or a graph with fewer than $$\ell$$ vertices. A graph is said to be (n,$$\ell)$$- connected if $$\kappa_{\ell}(G)\geq n$$. G. Chartrand, S. F. Kapoor, L. Lesniak and D. R. Lick [Bull. Bombay Math. Colloq. 2, 1-6 (1984)] proved that a graph of order p with independence number $$\beta$$ (G)$$\geq \ell \geq 2$$ is (n,$$\ell)$$-connected if the minimal degree $$\delta (G)\geq [p+(\ell -1)(n-2)]/\ell$$. Improving a result of J. Bondy [Studia Sci. Math. Hung. 4, 473-475 (1969; Zbl 0184.277)] the author states that a graph G of order $$p\geq 2$$ is (n,$$\ell)$$-connected if the degree-sequence $$d_ 1\leq...\leq d_ p$$ fulfills $$(d_ k\leq k+n- 2)\Rightarrow (d_{p-n+1}\geq p-k(\ell -1))$$ for each k with $$1\leq k\leq \lfloor (p-n+1)/\ell \rfloor$$. Also she gives a sufficient condition for a graph to contain at least n internally disjoint S-paths (S a set of $$\ell$$ vertices of G) in terms of the minimal degree using the theorem of Chartrand et al.
Reviewer: M.Hager

##### MSC:
 05C40 Connectivity
Full Text:
##### References:
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