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

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

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Graph Coloring Problems

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values.

Small Ramsey Numbers

A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. This paper studies the metric dimension of cartesian products $G\,\square\,H$. We prove that the metric dimension of $G\,\square\,G$ is tied in a strong sense to the minimum order of a so-called doubly resolving set in $G$. Using bounds on the order of doubly resolving sets, we establish bounds on $G\,\square\,H$ for many examples of $G$ and $H$. One of our main results is a family of graphs $G$ with bounded metric dimension for which the metric dimension of $G\,\square\,G$ is unbounded.

https://users.monash.edu.au/~davidwo/papers/MetricDimension-arXiv.pdf

On the Metric Dimension of Cartesian Products of Graphs

We study generators of metric spaces--sets of points with the property that every point of the space is uniquely determined by the distances from their elements. Such generators put a light on seemingly different kinds of problems in combinatorics that are not directly related to metric spaces. The two applications we present concern combinatorial search: problems on false coins known from the borderline of extremal combinatorics and information theory; and a problem known from combinatorial optimization--connected joins in graphs.We use results on the detection of false coins to approximate the metric dimension (minimum size of a generator for the metric space defined by the distances) of some particular graphs for which the problem was known and open. In the opposite direction, using metric generators, we show that the existence of connected joins in graphs can be solved in polynomial time, a problem asked in a survey paper of Frank. On the negative side we prove that the minimization of the number of components of a join is NP-hard.We further explore the metric dimension with some problems. The main problem we are led to is how to extend an isometry given on a metric generator of a metric space.

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On Metric Generators of Graphs

Resolvability in graphs and the metric dimension of a graph

Graphs & Digraphs masterfully employs student-friendly exposition, clear proofs, abundant examples, and numerous exercises to provide an essential understanding of the concepts, theorems, history, and applications of graph theory. Fully updated and thoughtfully reorganized to make reading and locating material easier for instructors and students, the Sixth Edition of this bestselling, classroom-tested text: Adds more than 160 new exercises Presents many new concepts, theorems, and examples Includes recent major contributions to long-standing conjectures such as the Hamiltonian Factorization Conjecture, 1-Factorization Conjecture, and Alspachs Conjecture on graph decompositions Supplies a proof of the perfect graph theorem Features a revised chapter on the probabilistic method in graph theory with many results integrated throughout the text At the end of the book are indices and lists of mathematicians names, terms, symbols, and useful references. There is also a section giving hints and solutions to all odd-numbered exercises. A complete solutions manual is available with qualifying course adoption. Graphs & Digraphs, Sixth Edition remains the consummate text for an advanced undergraduate level or introductory graduate level course or two-semester sequence on graph theory, exploring the subjects fascinating history while covering a host of interesting problems and diverse applications.

Graphs & Digraphs

On k-dimensional graphs and their bases

An edge-colored tree T is a rainbow tree if no two edges of T are assigned the same color. Let G be a nontrivial connected graph of order n and let k be an integer with 2 ≤ k ≤ n. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S ⊆ V(T). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G. For every two integers k and n ≥ 3 with 3 ≤ k ≤ n, the k-rainbow index of a unicyclic graph of order n is determined. For a set S of vertices in a connected graph G of order n, a collection lT1,T2,…,Ter of trees in G is said to be internally disjoint connecting S if these trees are pairwise edge-disjoint and V(Ti) ∩ V(Tj) = S for every pair i,j of distinct integers with 1 ≤ i,j ≤ e. For an integer k with 2 ≤ k ≤ n, the k-connectivity κk(G) of G is the greatest positive integer e for which G contains at least e internally disjoint trees connecting S for every set S of k vertices of G. It is shown that κk(Kn)=n-⌈k-2⌉ for every pair k,n of integers with 2 ≤ k ≤ n. For a nontrivial connected graph G of order n and for integers k and e with 2 ≤ k ≤ n and 1 ≤ e ≤ κk(G), the (k,e)-rainbow index rxk,e(G) of G is the minimum number of colors needed in an edge coloring of G such that G contains at least e internally disjoint rainbow trees connecting S for every set S of k vertices of G. The numbers rxk,e(Kn) are determined for all possible values k and e when n ≤ 6. It is also shown that for e e l1, 2r, rx3,e(Kn) = 3 for all n ≥ 6. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

Gary Chartrand

Papers

Resolvability in graphs and the metric dimension of a graph

Graphs & Digraphs

On k-dimensional graphs and their bases

Rainbow trees in graphs and generalized connectivity

The Point‐Arboricity of Planar Graphs