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Geoffrey Exoo

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Central European University

Anoop Chaturvedi

University of Allahabad

Armando Marques-Guedes

UNL - New University of Lisbon

Stephen Whittle

Manchester Metropolitan University

Iolanda Craifaleanu

Technical University Of Civil Engineering, Bucharest

Fabián Hoyos Patiño

Universidad Nacional de Colombia (National University of Colombia)

Roshan Chitrakar

Nepal College of Information Technology

Graduate Center of the City University of New York

University of Sarajevo

Paola Bacchetta

University of California, Berkeley

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Papers by Geoffrey Exoo

New Lower Bounds for 28 Classical Ramsey Numbers

Electronic Journal of Combinatorics, Jul 17, 2015

We establish new lower bounds for 28 classical two and three color Ramsey numbers, and describe t... more We establish new lower bounds for 28 classical two and three color Ramsey numbers, and describe the heuristic search procedures used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use (5, k)-colorings to obtain new (3, 3, k)-colorings, and (7, k)-colorings to obtain new (3, 4, k)-colorings. Some of the other new constructions in the paper are derived from two well known colorings: the Paley coloring of K 101 and the cubic coloring of K 127 . the electronic journal of combinatorics 22(3) (2015), #P3.11

The Hadwiger-Nelson problem with two forbidden distances

arXiv (Cornell University), May 15, 2018

In 1950 Edward Nelson asked the following simple-sounding question: How many colors are needed to... more In 1950 Edward Nelson asked the following simple-sounding question: How many colors are needed to color the Euclidean plane E 2 such that no two points distance 1 apart are identically colored? We say that 1 is a forbidden distance. For many years, we only knew that the answer was 4, 5, 6, or 7. In a recent breakthrough, de Grey [2] proved that at least five colors are necessary. In this paper we consider a related problem in which we require two forbidden distances, 1 and d. In other words, for a given positive number d = 1, how many colors are needed to color the plane such that no two points distance 1 or d apart are assigned the same color? We find several values of d, for which the answer to the previous question is at least 5. These results and graphs may be useful in constructing simpler 5-chromatic unit distance graphs.

A $6$-chromatic two-distance graph in the plane

arXiv: Combinatorics, 2019

We prove that if one colors each point of the Euclidean plane with one of five colors, then there... more

New Lower Bounds for 28 Classical Ramsey Numbers

The Electronic Journal of Combinatorics, 2015

We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe... more We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe the heuristic search procedures used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use $(5,k)$-colorings to obtain new $(3,3,k)$-colorings, and $(7,k)$-colorings to obtain new $(3,4,k)$-colorings. Some of the other new constructions in the paper are derived from two well known colorings: the Paley coloring of $K_{101}$ and the cubic coloring of $K_{127}$.

$Research paper thumbnail of On the Chromatic Number of $\mathbb{R}^n$ for Small Values of $n$$

On the Chromatic Number of $\mathbb{R}^n$ for Small Values of $n$

The lower bound for the chromatic number of $\mathbb{R}^n$ is improved for $n = 6, 7, 10, 11, 12,... more

$Research paper thumbnail of On the Chromatic Number of $$\mathbb {R}^4$$ R 4$

On the Chromatic Number of $$\mathbb {R}^4$$ R 4

Discrete & Computational Geometry, 2014

The lower bound for the chromatic number of R 4 is improved from 7 to 9. Three graphs with unit d... more The lower bound for the chromatic number of R 4 is improved from 7 to 9. Three graphs with unit distance embeddings in R 4 are described. The first is a 7-chromatic graph of order 14 whose chromatic number can be verified by inspection. The second is an 8-chromatic graph of order 26. In this case the chromatic number can be verified quickly by a simple computer program. The third graph is a 9-chromatic graph of order 65 for which computer verification takes about one minute.

Journal of Algebraic Combinatorics, Oct 4, 2012

A (k, g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order. We prese... more A (k, g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order. We present an explicit construction of (k, g)-Cayley graphs for all parameters k ≥ 2 and g ≥ 3 and generalize this construction to show that many wellknown small k-regular graphs of girth g can be constructed in this way. We also establish connections between this construction and topological graph theory, and address the question of the order of (k, g)-Cayley cages.

Bounds for the smallest k-chromatic graphs of given girth

ArXiv, 2019

Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We conside... more Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$. After giving an overview of what is known about $n_g(k)$, we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction of witnesses, and for the verification of their correctness. We also present the first examples of reasonably small order for $k = 4$ and $g > 5$. In particular, the new bounds include: $n_4(7) \leq 77$, $26 \leq n_6(4) \leq 66$, $30 \leq n_7(4) \leq 171$.

ε-Unit Distance Graphs

Discrete & Computational Geometry, Jun 4, 2004

We consider a variation on the problem of determining the chromatic number of the Euclidean plane... more We consider a variation on the problem of determining the chromatic number of the Euclidean plane and define the ε-unit distance graph to be the graph whose vertices are the points of E 2 , in which two points are adjacent whenever their distance is within ε of 1. For certain values of ε we are able to show that the chromatic number is exactly 7. For some smaller values we show the chromatic number is at least 5. We offer a conjecture, with some supporting evidence, that for any ε > 0 the chromatic number is 7.

On Some Small Classical Ramsey Numbers

The Electronic Journal of Combinatorics, 2013

This note is a report on a computer investigation of some small classical Ramsey numbers. We esta... more This note is a report on a computer investigation of some small classical Ramsey numbers. We establish new lower bounds for the classical Ramsey numbers $R(3,11)$ and $R(4,8)$. In the first case, the bound is improved from $46$ (a record that had stood for 46 years) to $47$; and in the second case the bound is improved from $57$ to $58$.

Some New Ramsey Colorings

The Electronic Journal of Combinatorics, 1998

New lower bounds for 15 classical Ramsey numbers are established. Several of the colorings are fo... more

Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

The Electronic Journal of Combinatorics, 2004

This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short ... more This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for $R_k(4)$ and $R_k(5)$ for some small $k$, including $415 \le R_3(5)$, $634 \le R_4(4)$, $2721 \le R_4(5)$, $3416 \le R_5(4)$ and $26082 \le R_5(5)$. Most of the new lower bounds are consequences of general constructions.

On the Ramsey Number $R(4,6)$

The Electronic Journal of Combinatorics, 2012

The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author ha... more The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author has found 37 new edge colorings of $K_{35}$ that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found using a heuristic search procedure.

A Simple Method for Constructing Small Cubic Graphs of Girths 14, 15 and 16

The Electronic Journal of Combinatorics, 1996

A method for constructing cubic graphs with girths in the range 13 to 16 is described. The method... more

The Chromatic Number of the Plane is At Least 5: A New Proof

Discrete & Computational Geometry, 2019

We present an alternate proof of the fact that given any 4-coloring of the plane there exist two ... more

On the defect of vertex-transitive graphs of given degree and diameter

by Geoffrey Exoo and Jozef Širáň

Journal of Combinatorial Theory, Series B, 2019

The Open University's repository of research publications and other research outputs On the defec... more

Locating-Dominating Codes in Cycles

The smallest cardinality of an r-locating-dominating code in a cycle Cn of length n is denoted by... more The smallest cardinality of an r-locating-dominating code in a cycle Cn of length n is denoted by MLDr (Cn). In this paper, we prove that for any r ≥ 5 and n ≥ nr when nr is large enough (nr = O(r3)) we have n/3 ≤ MLDr (Cn) ≤ n/3 + 1 if n ≡ 3 (mod 6) and MLD r (Cn) = [n/3] otherwise. Moreover, we determine the exact values of MLD3 (Cn) and MLD4 (Cn) for all n.

Improved bounds on binary identifying codes

2008 IEEE International Symposium on Information Theory, 2008

ABSTRACT The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in... more ABSTRACT The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Their motivation for identification came from finding malfunctioning processors in multiprocessor systems. Besides that identifying codes can also be applied to sensor networks. In this paper we consider identifying codes in Hamming spaces. We first concentrate on improving the lower bounds on codes which identify words within distance r &gt; 1. These improvements are achieved using a new approach. Then we proceed by introducing new lower bounds on codes identifying sets of words. Constructions for such codes with the best known cardinalities are also given.

Ranking measures for radially Moore graphs

Networks, 2010

... Joan Gimbert and Nacho López Departament de Matemàtica, Universitat de Lleida, Jaume II 69, 2... more

Covering and packing in graphs IV: Linear arboricity

Networks, 1981

ABSTRACT

New Lower Bounds for 28 Classical Ramsey Numbers

Electronic Journal of Combinatorics, Jul 17, 2015

We establish new lower bounds for 28 classical two and three color Ramsey numbers, and describe t... more We establish new lower bounds for 28 classical two and three color Ramsey numbers, and describe the heuristic search procedures used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use (5, k)-colorings to obtain new (3, 3, k)-colorings, and (7, k)-colorings to obtain new (3, 4, k)-colorings. Some of the other new constructions in the paper are derived from two well known colorings: the Paley coloring of K 101 and the cubic coloring of K 127 . the electronic journal of combinatorics 22(3) (2015), #P3.11

The Hadwiger-Nelson problem with two forbidden distances

arXiv (Cornell University), May 15, 2018

In 1950 Edward Nelson asked the following simple-sounding question: How many colors are needed to... more In 1950 Edward Nelson asked the following simple-sounding question: How many colors are needed to color the Euclidean plane E 2 such that no two points distance 1 apart are identically colored? We say that 1 is a forbidden distance. For many years, we only knew that the answer was 4, 5, 6, or 7. In a recent breakthrough, de Grey [2] proved that at least five colors are necessary. In this paper we consider a related problem in which we require two forbidden distances, 1 and d. In other words, for a given positive number d = 1, how many colors are needed to color the plane such that no two points distance 1 or d apart are assigned the same color? We find several values of d, for which the answer to the previous question is at least 5. These results and graphs may be useful in constructing simpler 5-chromatic unit distance graphs.

A $6$-chromatic two-distance graph in the plane

arXiv: Combinatorics, 2019

We prove that if one colors each point of the Euclidean plane with one of five colors, then there... more

New Lower Bounds for 28 Classical Ramsey Numbers

The Electronic Journal of Combinatorics, 2015

We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe... more We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe the heuristic search procedures used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use $(5,k)$-colorings to obtain new $(3,3,k)$-colorings, and $(7,k)$-colorings to obtain new $(3,4,k)$-colorings. Some of the other new constructions in the paper are derived from two well known colorings: the Paley coloring of $K_{101}$ and the cubic coloring of $K_{127}$.

$Research paper thumbnail of On the Chromatic Number of $\mathbb{R}^n$ for Small Values of $n$$

On the Chromatic Number of $\mathbb{R}^n$ for Small Values of $n$

The lower bound for the chromatic number of $\mathbb{R}^n$ is improved for $n = 6, 7, 10, 11, 12,... more

$Research paper thumbnail of On the Chromatic Number of $$\mathbb {R}^4$$ R 4$

On the Chromatic Number of $$\mathbb {R}^4$$ R 4

Discrete & Computational Geometry, 2014

The lower bound for the chromatic number of R 4 is improved from 7 to 9. Three graphs with unit d... more The lower bound for the chromatic number of R 4 is improved from 7 to 9. Three graphs with unit distance embeddings in R 4 are described. The first is a 7-chromatic graph of order 14 whose chromatic number can be verified by inspection. The second is an 8-chromatic graph of order 26. In this case the chromatic number can be verified quickly by a simple computer program. The third graph is a 9-chromatic graph of order 65 for which computer verification takes about one minute.

Journal of Algebraic Combinatorics, Oct 4, 2012

A (k, g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order. We prese... more A (k, g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order. We present an explicit construction of (k, g)-Cayley graphs for all parameters k ≥ 2 and g ≥ 3 and generalize this construction to show that many wellknown small k-regular graphs of girth g can be constructed in this way. We also establish connections between this construction and topological graph theory, and address the question of the order of (k, g)-Cayley cages.

Bounds for the smallest k-chromatic graphs of given girth

ArXiv, 2019

Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We conside... more Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$. After giving an overview of what is known about $n_g(k)$, we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction of witnesses, and for the verification of their correctness. We also present the first examples of reasonably small order for $k = 4$ and $g > 5$. In particular, the new bounds include: $n_4(7) \leq 77$, $26 \leq n_6(4) \leq 66$, $30 \leq n_7(4) \leq 171$.

ε-Unit Distance Graphs

Discrete & Computational Geometry, Jun 4, 2004

We consider a variation on the problem of determining the chromatic number of the Euclidean plane... more We consider a variation on the problem of determining the chromatic number of the Euclidean plane and define the ε-unit distance graph to be the graph whose vertices are the points of E 2 , in which two points are adjacent whenever their distance is within ε of 1. For certain values of ε we are able to show that the chromatic number is exactly 7. For some smaller values we show the chromatic number is at least 5. We offer a conjecture, with some supporting evidence, that for any ε > 0 the chromatic number is 7.

On Some Small Classical Ramsey Numbers

The Electronic Journal of Combinatorics, 2013

This note is a report on a computer investigation of some small classical Ramsey numbers. We esta... more This note is a report on a computer investigation of some small classical Ramsey numbers. We establish new lower bounds for the classical Ramsey numbers $R(3,11)$ and $R(4,8)$. In the first case, the bound is improved from $46$ (a record that had stood for 46 years) to $47$; and in the second case the bound is improved from $57$ to $58$.

Some New Ramsey Colorings

The Electronic Journal of Combinatorics, 1998

New lower bounds for 15 classical Ramsey numbers are established. Several of the colorings are fo... more

Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

The Electronic Journal of Combinatorics, 2004

This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short ... more This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for $R_k(4)$ and $R_k(5)$ for some small $k$, including $415 \le R_3(5)$, $634 \le R_4(4)$, $2721 \le R_4(5)$, $3416 \le R_5(4)$ and $26082 \le R_5(5)$. Most of the new lower bounds are consequences of general constructions.

On the Ramsey Number $R(4,6)$

The Electronic Journal of Combinatorics, 2012

The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author ha... more The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author has found 37 new edge colorings of $K_{35}$ that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found using a heuristic search procedure.

A Simple Method for Constructing Small Cubic Graphs of Girths 14, 15 and 16

The Electronic Journal of Combinatorics, 1996

A method for constructing cubic graphs with girths in the range 13 to 16 is described. The method... more

The Chromatic Number of the Plane is At Least 5: A New Proof

Discrete & Computational Geometry, 2019

We present an alternate proof of the fact that given any 4-coloring of the plane there exist two ... more

On the defect of vertex-transitive graphs of given degree and diameter

by Geoffrey Exoo and Jozef Širáň

Journal of Combinatorial Theory, Series B, 2019

The Open University's repository of research publications and other research outputs On the defec... more

Locating-Dominating Codes in Cycles

The smallest cardinality of an r-locating-dominating code in a cycle Cn of length n is denoted by... more The smallest cardinality of an r-locating-dominating code in a cycle Cn of length n is denoted by MLDr (Cn). In this paper, we prove that for any r ≥ 5 and n ≥ nr when nr is large enough (nr = O(r3)) we have n/3 ≤ MLDr (Cn) ≤ n/3 + 1 if n ≡ 3 (mod 6) and MLD r (Cn) = [n/3] otherwise. Moreover, we determine the exact values of MLD3 (Cn) and MLD4 (Cn) for all n.

Improved bounds on binary identifying codes

2008 IEEE International Symposium on Information Theory, 2008

ABSTRACT The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in... more ABSTRACT The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Their motivation for identification came from finding malfunctioning processors in multiprocessor systems. Besides that identifying codes can also be applied to sensor networks. In this paper we consider identifying codes in Hamming spaces. We first concentrate on improving the lower bounds on codes which identify words within distance r &gt; 1. These improvements are achieved using a new approach. Then we proceed by introducing new lower bounds on codes identifying sets of words. Constructions for such codes with the best known cardinalities are also given.

Ranking measures for radially Moore graphs

Networks, 2010

... Joan Gimbert and Nacho López Departament de Matemàtica, Universitat de Lleida, Jaume II 69, 2... more

Covering and packing in graphs IV: Linear arboricity

Networks, 1981

ABSTRACT