•Journal•ISSN: 1077-8926

# Electronic Journal of Combinatorics

About: Electronic Journal of Combinatorics is an academic journal. The journal publishes majorly in the area(s): Bipartite graph & Vertex (geometry). It has an ISSN identifier of 1077-8926. It is also open access. Over the lifetime, 4002 publication(s) have been published receiving 61559 citation(s). The journal is also known as: The Electronic Journal of Combinatorics.

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TL;DR: The On-Line Encyclopedia of Integer Sequences (OEIS) as mentioned in this paper is a database of 13,000 number sequences and is freely available on the Web (http://www.att.com/~njas/sequences/) and is widely used.

Abstract: The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences. It is freely available on the Web (http://www.research.att.com/~njas/sequences/) and is widely used.
There are several ways in which it benefits research: 1 It serves as a dictionary, to tell the user what is known about a particular sequence. There are hundreds of papers which thank the OEIS for assistance in this way.
1 The associated Sequence Fans mailing list is a worldwide network which has evolved into a powerful machine for tackling new problems.
1 As a direct source of new theorems, when a sequence arises in two different contexts.
1 As a source of new research, when one sees a sequence in the OEIS that cries out to be analyzed.
The 40-year history of the OEIS recapitulates the story of modern computing, from punched cards to the internet.
The talk will be illustrated with numerous examples, emphasizing new sequences that have arrived in the past few months. Many open problems will be mentioned.
Because of the profusion of books and journals, volunteers play an important role in maintaining the database. If you come across an interesting number sequence in a book, journal or web site, please send it and the reference to the OEIS. (You do not need to be the author of the sequence to do this.) There is a web site for sending in "Comments" or "New sequences".
Several new features have been added to the OEIS in the past year. Thanks to the work of Russ Cox, searches are now performed at high speed, and thanks to the work of Debby Swayne, there is a button which displays plots of each sequence. Finally, a "listen" button enables one to hear the sequence played on a musical instrument (try Recamaan's sequence A005132!).

4,548 citations

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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.

2,233 citations

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TL;DR: This work presents data which, to the best of its 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.

Abstract: 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.

548 citations

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TL;DR: The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter as mentioned in this paper, which is a largely unexplored area. But it is possible to obtain Moore-like upper bounds for the order of such graphs.

Abstract: The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'. This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

361 citations

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TL;DR: In this article, the authors extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a new class of labeled trees, which they call mobiles.

Abstract: We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.

296 citations