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17
List colouring squares of planar graphs
, 2008
"... In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most ⌊ 3 2 ∆ ⌋ + 1. We show that it is at most 3 2 ∆ (1 + o(1)), and indeed this is true for the list chromatic number and for more general classes of graphs. ..."
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Cited by 25 (5 self)
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In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most ⌊ 3 2 ∆ ⌋ + 1. We show that it is at most 3 2 ∆ (1 + o(1)), and indeed this is true for the list chromatic number and for more general classes of graphs.
A Comparison of Parallel Graph Coloring Algorithms
"... Dynamic irregular triangulated meshes are used in adaptive grid partial differential equation (PDE) solvers, and in simulations of random surface models of quantum gravity inphysics and cell membranes in biology. Parallel algorithms for random surface simulations and adaptive grid PDE solvers requir ..."
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Cited by 25 (0 self)
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Dynamic irregular triangulated meshes are used in adaptive grid partial differential equation (PDE) solvers, and in simulations of random surface models of quantum gravity inphysics and cell membranes in biology. Parallel algorithms for random surface simulations and adaptive grid PDE solvers require coloring of the triangulated mesh, so that neighboring vertices are not updated simultaneously. Graph coloring is also used in iterative parallel algorithms for solving large irregular sparse matrix equations. Here we introduce some parallel graph coloring algorithms based on wellknown sequential heuristic algorithms, and compare them with some existing parallel algorithms. These algorithms are implemented on both SIMD and MIMD parallel architectures and tested for speed, e ciency, and quality (the average number of colors required) for coloring random triangulated meshes and graphs from sparse matrix problems.
Improper choosability of graphs and maximum average degree
 IN JOURNAL OF GRAPH THEORY
, 2006
"... Improper choosability of planar graphs has been widely studied. In particular, ˇSkrekovski investigated the smallest integer gk such that every planar graph of girth at least gk is kimproper 2choosable. He proved [9] that 6 ≤ g1 ≤ 9; 5 ≤ g2 ≤ 7; 5 ≤ g3 ≤ 6 and ∀k ≥ 4, gk = 5. In this paper, we stu ..."
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Cited by 9 (2 self)
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Improper choosability of planar graphs has been widely studied. In particular, ˇSkrekovski investigated the smallest integer gk such that every planar graph of girth at least gk is kimproper 2choosable. He proved [9] that 6 ≤ g1 ≤ 9; 5 ≤ g2 ≤ 7; 5 ≤ g3 ≤ 6 and ∀k ≥ 4, gk = 5. In this paper, we study the greatest real M(k, l) such that every graph of maximum average degree less than M(k, l) is kimproper lchoosable. We prove that if l ≥ 2 then M(k, l) ≥ l + lk l+k. As a corollary, we deduce that g1 ≤ 8 and g2 ≤ 6, and we obtain new results for graphs of higher genus. We also provide an upper bound for M(k, l). This implies that for any fixed l, M(k, l) −−−→ k→ ∞ 2l.
Galleries and Light Matchings: Fat Cooperative Guards
 in Vision Geometry, Contemporary Mathematics
, 1991
"... For any collection of n disjoint line segments on the plane, such that no two are parallel and no three extensions meet in a common point, 3n/4 lights, none on any of the line segments, are occasionally necessary, and always sufficient to illuminate all points on all of the line segments. ..."
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Cited by 3 (0 self)
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For any collection of n disjoint line segments on the plane, such that no two are parallel and no three extensions meet in a common point, 3n/4 lights, none on any of the line segments, are occasionally necessary, and always sufficient to illuminate all points on all of the line segments.
The 5 Colour Theorem in Isabelle/Isar
 THEOREM PROVING IN HIGHER ORDER LOGICS, VOLUME 2410 OF LNCS
, 2002
"... Based on an inductive definition of triangulations, a theory of undirected planar graphs is developed in Isabelle/HOL. The proof of the 5 colour theorem is discussed in some detail, emphasizing the readability of the computer assisted proofs. ..."
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Cited by 3 (0 self)
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Based on an inductive definition of triangulations, a theory of undirected planar graphs is developed in Isabelle/HOL. The proof of the 5 colour theorem is discussed in some detail, emphasizing the readability of the computer assisted proofs.
The Complexity of the Four Colour Theorem
, 2009
"... The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a nontrivial computer verifica ..."
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Cited by 2 (0 self)
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The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a nontrivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq. In this paper we use the computational method for evaluating (in a uniform way) the complexity of mathematical problems presented in [8, 6] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem has roughly the same complexity as the Riemann hypothesis and almost four times the complexity of Fermat’s last theorem. 1
Circular chromatic number and graph minor, preprint
, 1997
"... Abstract. This paper proves that for any integer n ≥ 4 and any rational number r, 2 ≤ r ≤ n − 2, there exists a graph G which has circular chromatic number r and which does not contain Kn as a minor. 1. ..."
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Abstract. This paper proves that for any integer n ≥ 4 and any rational number r, 2 ≤ r ≤ n − 2, there exists a graph G which has circular chromatic number r and which does not contain Kn as a minor. 1.
Reasoning Using HigherOrder Abstract Syntax in a HigherOrder Logic Proof Environment: Improvements to Hybrid and a Case Study
"... is a joint program with Carleton University, administered by the Ottawa ..."
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Cited by 1 (1 self)
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is a joint program with Carleton University, administered by the Ottawa
Evaluating the Complexity of Mathematical Problems. Part 1
, 2009
"... In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of ..."
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In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of (very) simple programs. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied.