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Mirka Miller

Bio: Mirka Miller is an academic researcher from University of Minnesota. The author has contributed to research in topics: 1-planar graph & Bound graph. The author has an hindex of 3, co-authored 7 publications receiving 29 citations.

Papers
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01 Jan 1998
TL;DR: In this paper, the complete n-partite graph G = H m;n was considered and an optimal labeling of the vertices of G by distinct positive integers was given, where the vertex u and v are adjacent if and only if there exists a vertex u + v. This was the first known graph with this property.
Abstract: A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a finite number of isolated vertices to it will always yield a sum graph, and the sum number oe(G) of G is the smallest number of isolated vertices that will achieve this result. A labelling that realizes G + K oe(G) as a sum graph is said to be optimal. In this paper we consider G = H m;n , the complete n-partite graph on n 2 sets of m 2 nonadjacent vertices. We give an optimal labelling to show that oe(H 2;n ) = 4n \Gamma 5, and in the general case we give constructive proofs that oe(H m;n ) 2 \Omega\Gamma mn) and oe(H m;n ) 2 O(mn 2 ). We conjecture that oe(H m;n ) is asymptotically greater than mn, the cardinality of the vertex set; if so, then H m;n is the first known graph with this property. We also provide for the first time an optimal labelling of the complete bipatite graph Kmn whose smallest label is 1.

15 citations

01 Jan 2008
TL;DR: This paper considers super (a, d)edge-antimagic total labelings applied to the disjoint union of two stars K1,m and K2,n, a graph of order p and size q.
Abstract: Let G be a graph of order p and size q. An (a, d)-edge-antimagic total labeling of G is a one-to-one map f taking the vertices and edges onto 1, 2, . . . , p + q so that the edge-weights w(u, v) = f(u) + f(v) + f(uv), uv ∈ E(G), form an arithmetic progression, starting from a and having common difference d. Moreover, such a labeling is called super (a, d)edge-antimagic total if f(V (G)) = {1, 2, . . . , p}. This paper considers such labelings applied to a disjoint union of two stars K1,m and K1,n.

7 citations

Journal ArticleDOI
TL;DR: In this note, some of the open problems on various aspects of graph labelings which were posed by the participants during IWOGL 2009 and which have not been included in any of the other papers appearing in this volume are presented.

5 citations

01 Jan 2003
TL;DR: In this paper, the authors considered the disjoint union of graphs as sum graphs and provided an upper bound on the sum number of such graphs and an application for the exclusive sum number.
Abstract: In this paper we consider the disjoint union of graphs as sum graphs. We provide an upper bound on the sum number of a disjoint union of graphs and provide an application for the exclusive sum number of a graph. We conclude with some open problems.

2 citations

01 Jan 2017
TL;DR: This paper investigates the MaxDDBS problem when the host graph is a butterfly network and gives constructive lower bounds for subgraphs of maximum degree 4, 3 and 2.
Abstract: A maximum degree-diameter bounded subgraph problem can be seen as a degree-diameter problem restricted to certain host graphs. In this paper, we investigate the MaxDDBS problem when the host graph is a butterfly network. We give constructive lower bounds for subgraphs of maximum degree 4, 3 and 2. ∗ Was also at Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic. Deceased January 2016. N.H. BONG ET AL. /AUSTRALAS. J. COMBIN. 68 (2) (2017), 245–264 246

1 citations


Cited by
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Journal Article
TL;DR: In this survey I have collected everything I could find on graph labelings techniques that have appeared in journals that are not widely available.
Abstract: A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

2,367 citations

Book ChapterDOI
01 Jan 2013
TL;DR: An edge-magic total labeling on a graph G is a one-to-one map λ from V (G) \ cup E(G) onto the integers 1,2, …, v + e, where wt(xy) = k for any choice of edge xy.
Abstract: An edge-magic total labeling on a graph G is a one-to-one map λ from \( V (G) \ cup E(G) \) onto the integers 1,2, …, v + e, where v = | V (G) | and e = | E(G) |, with the property that, given any edge xy, $$\lambda (x) + \lambda (xy) + \lambda (y) = k$$ for some constant k. In other words, wt(xy) = k for any choice of edge xy. Then k is called the magic sum of G. Any graph with an edge-magic total labeling will be called edge-magic. described.

52 citations

Journal ArticleDOI
TL;DR: An optimal sum labelling scheme is provided for the generalised friendship graph, also known as the flower, and it is shown that its sum number is 2.

14 citations

Journal ArticleDOI
TL;DR: It is proved that the sum number of a hypertree (≔ connected, non-trivial and cycle-free hypergraph) is equal to 1, if a certain condition for the cardinalities of the edges is fulfilled.

10 citations

Book ChapterDOI
01 Jan 2013
TL;DR: A vertex-magic total labeling (VWC) as discussed by the authors is a one-to-one map of a graph from a vertex to the integers of the vertices of the graph, where the sum is over all vertices adjacent to the vertex x.
Abstract: A one-to-one map \(\lambda \) from \(E(G) \cup V (G)\) onto the integers \(\left \{1,2,\ldots,e + v\right \}\) is a vertex-magic total labeling if there is a constant h so that for every vertex x, $$\lambda (x) + \sum \lambda (xy) = h$$ (31) where the sum is over all vertices y adjacent to x So the magic requirement is wt(x) = h for all x The constant h is called the magic constant for λ Again, a graph with a vertex-magic total labeling will be called vertex-magic

10 citations