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### On a question by Erdős about edge-magic graphs

David Craft, Esther Hunt Tesar
1999 Discrete Mathematics
We give some general results about edge-magic graphs and show precisely which complete graphs are edge-magic.  ...  A (p; q) graph is edge-magic if the vertices and edges can be labeled with distinct elements from the set 1; 2; : : : ; p + q in such a way that the sum is the same along any edge.  ...  We will begin to answer Erdős' question by determining precisely which complete graphs are edge-magic.  ...

### Dense edge-magic graphs and thin additive bases

Oleg Pikhurko
2006 Discrete Mathematics
So, we also study s(k, n), motivated by the above connections to edge-magic graphs and by the fact that a few known functions from additive number theory can be expressed via s(k, n).  ...  For example, our estimate implies a new bound on the maximum size of quasi-Sidon sets, a problem posed by Erdős and Freud [On sums of a Sidon-sequence, J. Number Theory 38 (1991) 196-205].  ...  We construct an edge-magic graph G on [n] with m − n edges. Label i ∈ [n] by l(i) := a i . The magic sum will be s := k + m.  ...

### Dense Edge-Magic Graphs and Thin Additive Bases [article]

Oleg Pikhurko
2003 arXiv   pre-print
Also, applications to so-called edge-magic labellings of graphs are given.  ...  For example, our estimates of s(k,n) imply new bounds on the maximum size of quasi-Sidon sets, a problem posed by Erdos and Freud [J. Number Th.38 (1991) 196-205].  ...  If a n ≤ m, then M(n) ≥ m − n. Proof. We will construct an edge-magic graph G on [n] with m − n edges. Label i ∈ [n] by l(i) := a i . The magic sum will be s : = k + m.  ...

### Eighty Years of RamseyR(3, k)…and Counting! [chapter]

Joel Spencer
2011 Ramsey Theory
Decades go by without a hint of progress. What a delight when a problem is worked on over many many years with progress occurring incrementally until it finally succumbs.  ...  We deal throughout with graphs that are undirected and have neither loops nor multiple edges. We write G D .V; E/ where V; E are the sets of vertices and edges of G, respectively.  ...  By Erdős Magic there therefore existed such a triangle-free graph. Setting k D c 1 p n ln n this yielded R.3; k/ > n D c 2 k 2 ln 2 k, giving yet another new proof of Erdős 1961 gem.  ...

### Expand and Contract: Sampling graphs with given degrees and other combinatorial families [article]

James Y. Zhao
2013 arXiv   pre-print
The utility of the method is demonstrated via several examples, with particular emphasis on sampling labelled graphs with given degree sequence, a well-studied problem for which existing algorithms leave  ...  For graphs with given degrees, with maximum degree O(m^1/4) where m is the number of edges, we obtain an asymptotically uniform sample in O(m) steps, which substantially improves upon existing algorithms  ...  a multigraph and obtain a simple graph by removing loops and multiple edges.  ...

### Index

1999 Discrete Mathematics
Tesar, On a question by Erdo s about edge-magic graphs (Note) (1}3) 271}276 Flandrin, E., see M.E.K. Abderrezzak (1}3) 263}269 Fon-Der-Flaass, D.G., A.V. Kostochka and D.R.  ...  Katsuda, Isospectral graphs and isoperimetric constants (1}3) 33} 52 Hagedorn, T.R., On the existence of magic n-dimensional rectangles (1}3) 53} 63 Hagedorn, T.R., Magic rectangles revisited (1}3) 65}  ...

### Threshold phenomena in random structures

Joel E. Cohen
1988 Discrete Applied Mathematics
There are analogs of phase transition in the theory of random graphs, initiated by ErdGs and Renyi.  ...  This paper gives a nontechnical but precise account, without proofs, of some of the beautiful discoveries of Erdijs and RCnyi about threshold phenomena in graphs, describes an application of their methods  ...  Marois provided an opportunity to present an initial version of this paper at a conference of the Institut de la Vie held in Vienna, July 1979.  ...

### Embedding graphs into larger graphs: results, methods, and problems [article]

Miklós Simonovits, Endre Szemerédi
2019 arXiv   pre-print
Extremal Graph Theory is a very deep and wide area of modern combinatorics.  ...  We thank to several friends and colleagues the useful discussions about the many topics discussed in this survey.  ...  (d) Extremal graph theory optimizes on a Universe, and this Universe may be the family of Random Graphs. 63 Since a question of Erdős was answered by Babai, Simonovits, and Spencer [56] , (here Theorem  ...

### Reflections on Paul Erdős on His Birth Centenary

Krishnaswami Alladi, Steven Krantz, László Lovász, Vera T. Sós, Ronald L Graham, Joel Spencer, Jean-Pierre Kahane, Melvyn B. Nathanson
2015 Notices of the American Mathematical Society
This is Part I of a two-part feature on Paul Erdős following his centennial.  ...  There are eleven articles by leading experts who have reflected on the remarkable life, contributions, and influence of this towering figure of twentieth century mathematics.  ...  1 László Alpár , Pál Erdős (1913Erdős ( -1996, János Erőds, , Ervin Feldheim , Géza Grünwald (1913Grünwald ( -1944, Tibor Grünwald (Gallai) , Eszter Klein (1910Klein ( -1975, Dezső Lázár , György Szekeres  ...

### Magic and antimagic H-decompositions

N. Inayah, A. Lladó, J. Moragas
2012 Discrete Mathematics
By using the results on the sumset partition problem, we show that the complete graph K 2m+1 admits T -magic decompositions by any graceful tree with m edges.  ...  A decomposition of a graph G into isomorphic copies of a graph H is such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant.  ...  This work was supported by the Spanish Research Council under project MTM2008-06620-C03-01 and the Catalan Research Council under project 2009SGR01387.  ...

### Folk-Tale Networks: A Statistical Approach to Combinations of Tale Types

Julien D'Huy
2019 Journal of Ethnology and Folkloristics
The centrality of each type also seems to depend more on its age than on how widespread it is.  ...  The network of tale types seems to take the form of a 'small world' with a few types serving as bridges between highly connected sets of tale types.  ...  The result shows that a lot of distinct ATU tale types in the complete graph are adjacent, i.e. are connected by an edge.  ...

### Master index of volumes 121–130

1994 Discrete Mathematics
Massey, Incidence and strong edge colorings of graphs 122 (1993) 51-58 Bruen, A.A., Solution to a question of A.  ...  Beutelspacher on finite linear spaces 129 (1994) 219-220 Buekenhout, F., Questions about linear spaces 129 (1994) 19-27 Burlet, M., J.  ...

### Information-theoretic thresholds for community detection in sparse networks [article]

Jess Banks, Cristopher Moore
2016 arXiv   pre-print
random graphs with a certain effective degree.  ...  (Similar upper bounds were obtained independently by Abbe and Sandon.)  ...  This work was supported by the ARO under contract W911NF-12-R-0012 and the John Templeton Foundation.  ...

### Rigorous Result for the CHKNS Random Graph Model

Rick Durrett
2003 Discrete Mathematics & Theoretical Computer Science
International audience We study the phase transition in a random graph in which vertices and edges are added at constant rates.  ...  Here we sketch rigorous proofs for the first and third results and a new estimates about connectivity probabilities at the critical value.  ...  answer predates the question by more than 10 years.  ...

### Algorithms for channel assignment in mobile wireless networks using temporal coloring

Feng Yu, Amotz Bar-Noy, Prithwish Basu, Ram Ramanathan
2013 Proceedings of the 16th ACM international conference on Modeling, analysis & simulation of wireless and mobile systems - MSWiM '13
We model the problem of channel assignment in mobile networks as one of temporal coloring (T-coloring), that is, coloring a timevarying graph.  ...  We begin by analyzing two simple algorithms called SNAP and SMASH that take diametrically opposite positions on colors vs re-assignments, and provide theoretical results on the ranges of α in which one  ...  A open question is to find the best K for a given α.  ...
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