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.

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A Dynamic Survey of Graph Labeling

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.

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Edge-Magic Total Labelings

A sum labelling for the generalised friendship graph

Sum numbers of hypertrees

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

https://link.springer.com/content/pdf/10.1007%2F978-0-8176-8391-7_3.pdf

Vertex-Magic Total Labelings

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.

The sum number of the cocktail party graph

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.

/pdf/antimagic-labeling-of-the-union-of-stars-anjnvneclf.pdf

Antimagic labeling of the union of stars.

Some open problems on graph labelings

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.

The sum number of a disjoint union of graphs

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

/pdf/maxddbs-problem-on-butterfly-networks-2t1ozq61y5.pdf

Mirka Miller

Papers

The sum number of the cocktail party graph

Antimagic labeling of the union of stars.

Some open problems on graph labelings

The sum number of a disjoint union of graphs

MaxDDBS problem on butterfly networks.