We prove that every n-connected graph G of sufficiently large order contains a connected graph H on four vertices such that G Γ V Γ°H Γ is Γ°n Γ 3Γ-connected. This had been conjectured in Mader (High connectivity keeping sets in n-connected graphs, Combinatorica, to appear). Furthermore, we prove uppe
Critically (k, k)-connected graphs
β Scribed by Kiyoshi Ando; Yoko Usami
- Publisher
- Elsevier Science
- Year
- 1987
- Tongue
- English
- Weight
- 359 KB
- Volume
- 66
- Category
- Article
- ISSN
- 0012-365X
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A graph G which iit n-connected (but not (I! I)-connected) is defined ro be k-xitical if for every S 6; V(G), where f S i d k. the connectivity of G -I S is h -/S ia We will say that G is an (n\*,k\*) graph if G is n-conneckxt (b:lt nat (n t Itconnected) and k-crirical (hut not (k c l)criticaf). Thi
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A noncomplete graph G is called an (n, k)-graph if it is n-connected and G&X is not (n&|X | +1)-connected for any X V(G) with |X | k. Mader conjectured that for k 3 the graph K 2k+2 -(1-factor) is the unique (2k, k)-graph. We settle this conjecture for k 4.