https://nova.newcastle.edu.au/vital/access/ /manager/Index en-au 5 https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:22237 G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V(G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n²) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.]]> Tue 04 Feb 2020 10:56:52 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:18443 Sat 24 Mar 2018 07:59:47 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:16253 Sat 24 Mar 2018 07:54:13 AEDT ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:32758 sum graph G is a graph with an injective mapping of the vertex set of G onto a set of positive integers S in such a way that two vertices of G are adjacent if and only if the sum of their labels is an element of S. In an exclusive sum graph the integers of S that are the sum of two other integers of S form a set of integers that label a collection of isolated vertices associated with the graph G. A graph bears a k-exclusive sum labelling (abbreviated k-ESL), if the set of isolated vertices is of cardinality k, an optimal exclusive sum labelling, if k is as small as possible, and Δ-optimal if k equals the maximum degree of the graph. In this paper, observing that the property of having a k-ESL is hereditary, we provide a characterisation of graphs that have a k-exclusive sum labelling, for any k ≥ 1, in terms of describing a universal graph for the property.]]> Mon 23 Jul 2018 11:04:20 AEST ]]> https://nova.newcastle.edu.au/vital/access/ /manager/Repository/uon:47958 Mon 13 Feb 2023 14:27:59 AEDT ]]>