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

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].

The maximum clique problem

http://compalg.inf.elte.hu/~tony/Kutatas/BinPacking/Johnson-BinPacking-NP-Column-JAlg-2002.pdf

The NP-completeness column: An ongoing guide

Minimization with orthogonality constraints (e.g., $$X^\top X = I$$) and/or spherical constraints (e.g., $$\Vert x\Vert _2 = 1$$) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform--a Crank-Nicolson-like update scheme--to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 % to the best known solution on the largest problem "tai256c" in QAPLIB can be reached in 5 min on a typical laptop.

/pdf/a-feasible-method-for-optimization-with-orthogonality-28mr8i16hg.pdf

A feasible method for optimization with orthogonality constraints

An $L(2,1)$-labeling of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $d(x,y)=1$ and $|f(x)-f(y)|\ge 1$ if $d(x,y)=2$. The $L(2,1)$-labeling number $\lambda(G)$ of $G$ is the smallest number $k$ such that $G$ has an $L(2,1)$-labeling with $\max\{f(v):v\in V(G)\}=k$. In this paper, we give exact formulas of $\lambda(G\cup H)$ and $\lambda(G+H)$. We also prove that $\lambda(G)\le\Delta^2+\Delta$ for any graph $G$ of maximum degree $\Delta$. For odd-sun-free (OSF)-chordal graphs, the upper bound can be reduced to $\lambda(G)\le 2\Delta+1$. For sun-free (SF)-chordal graphs, the upper bound can be reduced to $\lambda(G)\le \Delta+2\chi(G)-2$. Finally, we present a polynomial time algorithm to determine $\lambda(T)$ for a tree $T$.

https://ir.nctu.edu.tw:443/bitstream/11536/1295/1/A1996UL33300013.pdf

The $L(2,1)$-Labeling Problem on Graphs

The k-domination problem is to find a minimum cardinality vertex set D of a graph such that every vertex of the graph is within distance k from some vertex of D, where k is a positive integer. The k-stability problem is to find a maximum cardinality vertex set S such that the distance between any two distinct vertices of S is greater than k. For sun-free chordal graphs, $2k$-stability and k-domination are dual problems. In particular, a minimum cardinality set of vertices D such that every vertex is within distance k of D has the same cardinality as a maximum cardinality set of vertices S such that the distance between every pair of vertices in S is greater than $2k$. To obtain this result we establish some theorems about the powers and radius of chordal graphs. Efficient algorithms for both problems on sun-free chordal graphs are obtained by transforming them to solvable cases of the clique covering and vertex packing problems. We also prove the NP-completeness of both problems on bipartite and chordal g...

The k-Domination and k-Stability Problems on Sun-Free Chordal Graphs

https://ir.nctu.edu.tw:443/bitstream/11536/30465/1/000087401100005.pdf

On L(d, 1) -labelings of graphs

Note: Power domination in graphs

List of Figures. Preface. 1. Introduction. 2. Computation of Reliability. 3. Design of Optimal Consecutive Systems. 4. The Lifetime Distribution. 5. Asymptotic Analysis. 6. Window Systems. 7. The Network Model. 8. Consecutive-2 Graphs. 9. Some Related Systems. 10. Applications. References. Index.

Gerard J. Chang

Papers

The $L(2,1)$-Labeling Problem on Graphs

The k-Domination and k-Stability Problems on Sun-Free Chordal Graphs

On L(d, 1) -labelings of graphs

Note: Power domination in graphs

Reliabilities of Consecutive-k Systems