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### A solution to a colouring problem of P. Erdős

Herbert Fleischner, Michael Stiebitz
1992 Discrete Mathematics
Introduction The purpose of this paper is to prove Theorem 1.1 which provides an affirmative solution to a colouring problem posed by P.  ...  For a given graph G, x(G) is the minimum number of colours needed to colour the vertices of G such that adjacent vertices have distinct colours.  ...

### The Erdös-Lovász Tihany conjecture and complete minors

Ken-Ichi Kawarabayashi, Anders Sune Pedersen, Bjarne Toft
2011 Journal of Combinatorics
In this paper we settle this latter problem for a few small additional values of s and t.  ...  If Hadwiger's Conjecture holds, then this latter problem might be easier to settle than the Erdős-Lovász Tihany Conjecture.  ...  Nevertheless, we expect Problem 3.1 to be very difficult, since a positive solution of Problem 3.1 restricted to 6-chromatic graphs would imply a positive solution to the Double-Critical Graph Conjecture  ...

### 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.  ...  Some results discussed here got stronger emphasis, since they are connected to Lov\'asz (and sometimes to us).  ...  We thank above all to József Balogh and András Gyárfás, and also to Zoltán Füredi, János Pach, Jan Hladký, Zoltán L.  ...

### A Parallel Branch and Bound Algorithm for the Maximum Labelled Clique Problem [article]

Ciaran McCreesh, Patrick Prosser
2014 arXiv   pre-print
The maximum labelled clique problem is a variant of the maximum clique problem where edges in the graph are given labels, and we are not allowed to use more than a certain number of distinct labels in  ...  a solution.  ...  and from P 22 colourOrder :: (Vertex Set P ) → (Vertex Array, Int Array) 23 begin 24 (order , bounds ) ← ([], colourable = ∅ do 30 v ← the first vertex of colourable 31 append v to order , and colour to  ...

### A parallel branch and bound algorithm for the maximum labelled clique problem

Ciaran McCreesh, Patrick Prosser
2014 Optimization Letters
The maximum labelled clique problem is a variant of the maximum clique problem where edges in the graph are given labels, and we are not allowed to use more than a certain number of distinct labels in  ...  a solution.  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s)  ...

### Some solved and unsolved problems in combinatorial number theory, ii

P. Erdős, A. Sárközy
1993 Colloquium Mathematicum
called a k-partition (or k-colouring) of S, and the subsets A 1 , . . . , A k are referred to as classes.  ...  integer, then let f (N, A, t) denote the number of solutions of N i=1 ε i a i = t where ε i = 0 or 1 .  ...

### On zero-sum turan problems of Bialostocki and Dierker

Y. Caro, Y. Roditty
1992 Journal of the Australian Mathematical Society
A Z k -colouring of a graph G is a colouring of the edges of G by Z k , the additive group of integers modulo k , avoiding a copy of a given graph H, for which the sum of the values on its edges is 0 (  ...  By the Zero-Sum Turan number, denoted T(n, G, Z k ), k | m , we mean the maximum number of edges in a Z k -colouring of a graph on n vertices that contains no zero-sum (mod k) copy of G .  ...  Acknowledgement We would like to thank the referee for his helpful remarks. On zero-sum problems  ...

### Paul Erdős's (1913-1996)

András Sárközy
1997 Acta Arithmetica
As an answer to this question, I soon published (in the Acta Arithmetica) my first paper based on an Erdős problem.  ...  However, when after a good sleep you awoke next morning your head was full of his problems and ideas, and you were just unable to think about anything else; and in at most two days you were longing again  ...  for the solution of a numbertheoretic problem.  ...

### A combinatorial mathematician in Norway: some personal reflections

Helge Tverberg
2001 Discrete Mathematics
In the paper, I ÿrst try to give some impression of Norwegian contributions to combinatorics in the 20th century. This is followed by some remarks on my own combinatorial experiences.  ...  Later, I was to meet him quite often at conferences. I even checked a solution by David Preiss of one of Erdős' prize problems when we were in Durham, UK, in 1974.  ...  Secondly, he notes that for p ≡ 1 (mod 4) there are solutions of the congru-ence a 2 ≡ −1 (mod p), as follows from the well-known fact-Wilson's theorem-that (p − 1)! ≡ −1 (mod p) for all primes p.  ...

### Solved and Unsolved Problems

Michael Rassias
I Six new problems -solutions solicited Solutions will appear in a subsequent issue. 179. Let p = p 1 p 2 · · · p n and q = q 1 q 2 · · · q n be two permutations.  ...  Does an n-universal word of length n 2 − 2n + 4 exist? 185* (Erdős' unit distance problem).  ...  Therefore We also solicit your new problems with their solutions for the next "Solved and Unsolved Problems" column, which will be devoted to Fundamentals of Mathematical Analysis.  ...

### The Erdős–Rothschild problem on edge-colourings with forbidden monochromatic cliques

OLEG PIKHURKO, KATHERINE STADEN, ZELEALEM B. YILMA
2017 Mathematical proceedings of the Cambridge Philosophical Society (Print)
Our final result is a stability theorem for complete multipartite graphs G, describing the asymptotic structure of such G with F(G; k ) = F(n; k ) · 2 o(n 2) in terms of solutions to the optimisation problem  ...  This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases.  ...  Erdős and Rothschild also considered the generalisation of the problem, where one forbids a monochromatic graph H (the same for each colour).  ...

### Clique Is Hard on Average for Regular Resolution [article]

Albert Atserias, Ilario Bonacina, Susanna F. de Rezende, Massimo Lauria, Jakob Nordström, Alexander Razborov
2020 arXiv   pre-print
We prove that for k ≪√(n) regular resolution requires length n^Ω(k) to establish that an Erdős-Rényi graph with appropriately chosen edge density does not contain a k-clique.  ...  This lower bound is optimal up to the multiplicative constant in the exponent, and also implies unconditional n^Ω(k) lower bounds on running time for several state-of-the-art algorithms for finding maximum  ...  Acknowledgements This work has been a long journey, and different subsets of the authors want to acknowledge fruitful and enlightening discussions with different subsets from the following list of colleagues  ...

### Sums and products along sparse graphs [article]

Noga Alon, Omer Angel, Itai Benjamini, Eyal Lubetzky
2009 arXiv   pre-print
n^1/2+δ for its sum-product over the integers implies a lower bound of n^1+δ for the original Erdős-Szemerédi problem.  ...  A key element in our proofs is a reduction from the sum-product of a matching to the maximum number of translates of a set of integers into the perfect squares.  ...  This work was initiated while the second and third authors were visiting the Theory Group of Microsoft Research.  ...

### Paul Erdos. 26 March 1913 -- 20 September 1996: Elected For.Mem.R.S. 1989

A. Baker, B. Bollobas
1999 Biographical Memoirs of Fellows of the Royal Society
Szeg , P. Turán and perhaps, above all, the subject of this memoir, Pál (Paul) Erdos.  ...  As Ernst Straus put it, Erdos was 'the crown prince of problem solvers and the undisputed monarch of problem posers'.  ...  logn 1 2 n + o(n) f (m) ∑ p f (p) 2 /p = ∞ x ∈r lim n→ ∞ A x (n)/n = F (x), f ′(p) = f (p) | f (p)| ≤ 1 f ′(p) = 1 ∑ f (p)≠0 1/p A problem of a rather different nature also occupied Erdos for several  ...

### Structural and Logical Approaches to the Graph Isomorphism Problem [chapter]

Martin Grohe
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
To use colour refinement as an isomorphism test, apply it to disjoint union of the input graphs G , H.Colour refinement distinguishes G and H if there is a colour c of the stable colouring such that G  ...  Theorem (Tinhofer 1991) Colour refinement distinguishes G and H iff the linear program has no solution.  ...  Level-k Sherali-Adams relaxation (L k ) Variables Otto 2012) k-WL distinguishes G and H iff COMP k−1 ∪ CONT k ∪ NN k has a solution.  ...
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