/manager/Index ${session.getAttribute("locale")} 5 Distance-locally disconnected graphs /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 ]]> On the uniqueness of D-vertex magic constant /manager/Repository/uon:20696 D(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, Σu∈ND(v) f(u) is a constant, called D-vertex magic constant. O'Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.]]> Sat 24 Mar 2018 07:55:40 AEDT ]]> A note on the locating-total domination in graphs /manager/Repository/uon:32759 Mon 23 Jul 2018 11:04:20 AEST ]]>