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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Further results on sequentially additive graphs

100%
Hegde S., Miller M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 2
251-268
EN
Given a graph G with p vertices, q edges and a positive integer k, a k-sequentially additive labeling of G is an assignment of distinct numbers k,k+1,k+2,...,k+p+q-1 to the p+q elements of G so that every edge uv of G receives the sum of the numbers assigned to the vertices u and v. A graph which admits such an assignment to its elements is called a k-sequentially additive graph. In this paper, we give an upper bound for k with respect to which the given graph may possibly be k-sequentially additive using the independence number of the graph. Also, we prove a variety of results on k-sequentially additive graphs, including the number of isolated vertices to be added to a complete graph with four or more vertices to be simply sequentially additive and a construction of an infinite family of k-sequentially additive graphs from a given k-sequentially additive graph.
2
Content available remote

Distance-Locally Disconnected Graphs

81%
Miller M., Ryan J., Ryjáček Z.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 1
203-215
EN
For an integer k ≥ 1, we say that a (finite simple undirected) graph 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.
3
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Survey of certain valuations of graphs

71%
Bača M., MacDougall J., Miller M., Slamin, Wallis W.
Discussiones Mathematicae Graph Theory
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2000
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tom 20
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nr 2
219-229
EN
The study of valuations of graphs is a relatively young part of graph theory. In this article we survey what is known about certain graph valuations, that is, labeling methods: antimagic labelings, edge-magic total labelings and vertex-magic total labelings.
4
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A Note on the Locating-Total Domination in Graphs

71%
Miller M., Rajan R. S., Jayagopal R., Rajasingh I., Manuel P.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 3
745-754
EN
In this paper we obtain a sharp (improved) lower bound on the locating-total domination number of a graph, and show that the decision problem for the locating-total domination is NP-complete.
5
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Vertex-antimagic total labelings of graphs

61%
Bača M., MacDougall J., Bertault F., Miller M., Simanjuntak R., Slamin
Discussiones Mathematicae Graph Theory
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2003
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tom 23
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nr 1
67-83
EN
Open problem 1. For the paths Pₙ and the cycles Cₙ, determine if there is a vertex-antimagic total labeling for every feasible pair (a,d). Open problem 2. Apart from duality, how can a vertex-antimagic total labeling for a graph be used to construct another vertex-antimagic total labeling for the same graph, preferably with different a and d? Open problem 3. In Theorem 3, we found a way to construct VATL for a graph G from a vertex-magic total labeling of G. Are there other ways to do this? Open problem 4. Find, if possible, some structural characteristics of a graph which make a vertex-antimagic total labeling impossible
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