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63
Models of Random Regular Graphs
 In Surveys in combinatorics
, 1999
"... In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular g ..."
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Cited by 225 (33 self)
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In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right hand sides. 1
Restricted colorings of graphs
 in Surveys in Combinatorics 1993, London Math. Soc. Lecture Notes Series 187
, 1993
"... The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, al ..."
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Cited by 97 (17 self)
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The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems. This is mainly a survey of recent and less recent results in the area, but it contains several new results as well.
A parallel algorithmic version of the local lemma
, 1991
"... The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for ..."
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Cited by 75 (10 self)
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The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.
Graph Treewidth and Geometric Thickness Parameters
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restri ..."
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Cited by 22 (9 self)
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Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and stararboricity.
Partitioning into graphs with only small components
 J. Combin. Theory Ser. B
, 2003
"... Abstract. The paper presents several results on edge partitions and vertex partitions of graphs into graphs with bounded size components. We show that every graph of bounded treewidth and bounded maximum degree admits such partitions. We also show that an arbitrary graph of maximum degree four has ..."
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Cited by 21 (0 self)
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Abstract. The paper presents several results on edge partitions and vertex partitions of graphs into graphs with bounded size components. We show that every graph of bounded treewidth and bounded maximum degree admits such partitions. We also show that an arbitrary graph of maximum degree four has a vertex partition into two graphs, each of which has components on at most 57 vertices. Some generalizations of the last result are also discussed. 1.
Acyclic colourings of planar graphs with large girth
 J. London Math. Soc
, 1999
"... A proper vertexcolouring of a graph is acyclic if there are no 2coloured cycles. It is known that every planar graph is acyclically 5colourable, and that there are planar graphs with acyclic chromatic number χ a � 5 and girth g � 4. It is proved here that a planar graph satisfies χ ..."
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Cited by 17 (0 self)
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A proper vertexcolouring of a graph is acyclic if there are no 2coloured cycles. It is known that every planar graph is acyclically 5colourable, and that there are planar graphs with acyclic chromatic number χ a � 5 and girth g � 4. It is proved here that a planar graph satisfies χ
Polynomial treewidth forces a large gridlikeminor
, 2008
"... Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ℓ × ℓ grid minor is exponential in ℓ. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. ..."
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Cited by 12 (2 self)
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Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ℓ × ℓ grid minor is exponential in ℓ. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A gridlikeminor of order ℓ in a graph G is a set of paths in G whose intersection graph is bipartite and contains a Kℓminor. For example, the rows and columns of the ℓ × ℓ grid are a gridlikeminor of order ℓ + 1. We prove that polynomial treewidth forces a large gridlikeminor. In particular, every graph with treewidth at least cℓ 4 √ log ℓ has a gridlikeminor of order ℓ. As an application of this result, we prove that the cartesian product G □ K2 contains a Kℓminor whenever G has treewidth at least cℓ 4 √ log ℓ.
Acyclic Colourings of 1Planar Graphs
, 1999
"... . A graph is 1planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1planar graph is at most 20. Keywords. Acyclic colouring, planar graphs, 1planar graphs. 1 ..."
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Cited by 11 (0 self)
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. A graph is 1planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1planar graph is at most 20. Keywords. Acyclic colouring, planar graphs, 1planar graphs. 1
Linear Arboricity and Linear KArboricity of Regular Graphs
 Graphs Combin
, 2001
"... We nd upper bounds on the linear karboricity of dregular graphs using a probabilistic argument. For small k these bounds are new. For large k they blend into the known upper bounds on the linear arboricity of regular graphs. 1 Introduction A linear forest is a forest each of whose components i ..."
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We nd upper bounds on the linear karboricity of dregular graphs using a probabilistic argument. For small k these bounds are new. For large k they blend into the known upper bounds on the linear arboricity of regular graphs. 1 Introduction A linear forest is a forest each of whose components is a path. The linear arboricity of a graph G is the minimum number of linear forests required to partition E(G) and is denoted by la(G). It was shown by Akiyama, Exoo and Harary [1] that la(G) = 2 when G is cubic, and they conjectured that every dregular graph has linear arboricity exactly d(d + 1)=2e. This was shown to be asymptotically correct as d !1 in [3], and in [4] the following result is shown. Theorem 1 There is an absolute constant c > 0 such that for every dregular graph G la(G) d 2 + cd 2=3 (log d) 1=3 : (Actually a slightly weaker result is proved explicitly there, but it is noted that the same proof with a little more care gives this theorem.) A linear kforest ...