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Given a function f : N → R, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f (k) vertices whenever k ≤ (n − f (k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is(More)
In the class of k-connected claw-free graphs, we study the stability of some hamilto-nian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e., for any of these properties there is an innnite family of graphs G k of(More)
We introduce the concept of an edge-colouring total k-labelling. This is a labelling of the vertices and the edges of a graph G with labels 1, 2,. .. , k such that the weights of the edges define a proper edge colouring of G. Here the weight of an edge is the sum of its label and the labels of its two endvertices. We define χ ′ t (G) to be the smallest(More)
In generalizing the concept of a pancyclic graph, we say that a graph is 'weakly pancyclic' if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it(More)
Environmental stresses commonly encountered by plants lead to rapid transient elevations in cytosolic free calcium concentration ([Ca2+]cyt) (Bush, 1995; Knight et al., 1991). These cellular calcium (Ca2+) signals lead ultimately to the increased expression of stress-responsive genes, including those encoding proteins of protective function (Knight et al.,(More)