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

On irregular total labellings

Let $$G=(V,E)$$G=(V,E) be a connected graph with $$\left| V \right| =n$$V=n and $$\left| E \right| = m.$$E=m. A bijection $$f:E \rightarrow \{1,2, \dots , m\}$$f:E?{1,2,?,m} is called a local antimagic labeling if for any two adjacent vertices u and v, $$w(u)\ne w(v),$$w(u)?w(v), where $$w(u)=\sum \nolimits _{e\in E(u)}{f(e)},$$w(u)=?e?E(u)f(e), and E(u) is the set of edges incident to u. Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the color w(v). The local antimagic chromatic number $$\chi _{la}(G)$$?la(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper we present several basic results on this new parameter.

Local Antimagic Vertex Coloring of a Graph

A total edge-irregular k-labelling ξ : V (G) ∪ E(G) → {1, 2, . . . , k} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every tree T of maximum degree ∆ on p vertices tes(T ) = max{d(p + 1)/3e, d(∆ + 1)/2e}.

http://www.discuss.wmie.uz.zgora.pl/php/discuss3.php?ip=&url=pdf&nIdA=16147

Total edge irregularity strength of trees

On the total vertex irregularity strength of trees

Magic squares are among the more popular mathematical recreations. Over the last 50 years, many generalizations of magic ideas have been applied to graphs. Recently there has been a resurgence of interest in magic labelings due to a number of results that have applications to the problem of decomposing graphs into trees. Key features of this second edition include: a new chapter on magic labeling of directed graphs applications of theorems from graph theory and interesting counting arguments new research problems and exercises covering a range of difficulties a fully updated bibliography and index This concise, self-contained exposition is unique in its focus on the theory of magic graphs/labelings. It may serve as a graduate or advanced undergraduate text for courses in mathematics or computer science, and as reference for the researcher.

Alison M. Marr

Papers

Magic Graphs