General connectivity

Abstract

Let X be a finite structure having property P, written X e P, and let Y be a nonempty subset of X. Then the general connectivity κ (X,Y : P) is the minimum cardinality of Z ⊂ Y such that X - Z ∉ P. Clearly graph connectivity (resp. line-connectivity) is the special case X=G=(V,E), Y=V(resp. E), and P=connected. Illustrations for groups, numbers, and graphs are given, including the following:(a) X=Y is a finite group and P means "generates X"; this case of general connectivity is easily characterized. (b) X=Y=Nn={1,2,...,n} and P means "contains a k-term arithmetic progression", as suggested by van der Waerden's Theorem. (c) X=Y=Nn again, but now P means that there exist three numbers x,y,z such that x+y=z. When x=y is permitted, this is reminiscent of Schur's existence theorem; otherwise Rado's theorem. (d) X=Kp, Y=E(Kp) and P means "contains Kn with n≦p". This is a reformulation of Turán's original problem in extremal graph theory. (e) X=G=(V,E), Y=E and P=hamiltonian. An extremal problem of this type was solved by Ore. (f) X=G is a connected graph, P=not graceful, and Y=V. (g) Again X=G and P=not graceful, but now Y=E. These graceful connectivities always exist, provided it is a true conjecture that all trees are graceful.