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21
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 31 (19 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
Graphs and partially ordered sets: recent results and new directions
 JACOBSON (EDS.), SURVEYS IN GRAPH THEORY, CONGRESSUS NUMERANTIUM
, 1996
"... We survey some recent research progress on topics linking graphs and finite partially ordered sets. Among these topics are planar graphs, hamiltonian cycles and paths, graph and hypergraph coloring, online algorithms, intersection graphs, inclusion orders, random methods and ramsey theory. In each ..."
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Cited by 9 (2 self)
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We survey some recent research progress on topics linking graphs and finite partially ordered sets. Among these topics are planar graphs, hamiltonian cycles and paths, graph and hypergraph coloring, online algorithms, intersection graphs, inclusion orders, random methods and ramsey theory. In each case, we discuss open problems and future research directions.
New perspectives on interval orders and interval graphs
 in Surveys in Combinatorics
, 1997
"... Abstract. Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the so ..."
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Cited by 7 (5 self)
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Abstract. Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the social sciences have investigated structural, algorithmic, enumerative, combinatorial, extremal and even experimental problems associated with them. In this article, we survey recent work on interval orders and interval graphs, including research on online coloring, dimension estimates, fractional parameters, balancing pairs, hamiltonian paths, ramsey theory, extremal problems and tolerance orders. We provide an outline of the arguments for many of these results, especially those which seem to have a wide range of potential applications. Also, we provide short proofs of some of the more classical results on interval orders and interval graphs. Our goal is to provide fresh insights into the current status of research in this area while suggesting new perspectives and directions for the future. 1.
Coloring intersection graphs of geometric figures, in: Towards a Theory of Geometric Graphs
 Contemporary Mathematics 342, Amer. Math. Soc
, 2004
"... given clique number ..."
Vertex Colouring and Forbidden Subgraphs  a Survey
, 2003
"... There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs de ned in terms of forbidden induced subgraph conditions. ..."
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Cited by 5 (0 self)
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There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs de ned in terms of forbidden induced subgraph conditions.
On Visibility and Covering By Convex Sets
, 1999
"... A set X ` IR d is nconvex if among any n its points there exist two such that the segment connecting them is contained in X. Perles and Shelah have shown that any closed (n + 1)convex set in the plane is the union of at most n 6 convex sets. We improve their bound to 18n 3 , and show a lower ..."
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Cited by 4 (0 self)
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A set X ` IR d is nconvex if among any n its points there exist two such that the segment connecting them is contained in X. Perles and Shelah have shown that any closed (n + 1)convex set in the plane is the union of at most n 6 convex sets. We improve their bound to 18n 3 , and show a lower bound of order \Omega\Gamma n 2 ). We also show that if X ` IR 2 is an nconvex set such that its complement has onepoint pathconnectivity components, ! 1, then X is the union of O(n 4 + n 2 ) convex sets. Two other results on the nconvex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4). 1 Introduction 1.1 Review of results Let X be a set in the ddimensional Euclidean space IR d . Let fl(X) denote the minimum (cardinal) number k such that X can be expressed as the union of k convex sets (not necessarily disjoint ones). For a subset Y ` X, we write fl X (Y ) for the minimum number of convex subsets of X whose union contains Y . We investigate the ...
THE MAXIMUM NUMBER OF EDGES IN 2K_2FREE GRAPHS OF BOUNDED DEGREE
, 1990
"... A graph is 2K,free if it does not contain an independent pair of edges as an induced subgraph. We show that if G is 2K,free and has maximum degree A(G) = D, then G has at most 5D2/4 edges if D is even. If D is odd, this bound can be improved to (5D * 20 + 1)/4. The extremal graphs are unique. ..."
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Cited by 4 (0 self)
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A graph is 2K,free if it does not contain an independent pair of edges as an induced subgraph. We show that if G is 2K,free and has maximum degree A(G) = D, then G has at most 5D2/4 edges if D is even. If D is odd, this bound can be improved to (5D * 20 + 1)/4. The extremal graphs are unique.
INDUCED CYCLES AND CHROMATIC NUMBER
"... Abstract. We prove that, for any pair of integers k, l ≥ 1, there exists an integer N(k, l) such that every graph with chromatic number at least N(k, l) contains either Kk or an induced odd cycle of length at least 5 or an induced cycle of length at least l. Given a graph with large chromatic number ..."
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Cited by 2 (2 self)
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Abstract. We prove that, for any pair of integers k, l ≥ 1, there exists an integer N(k, l) such that every graph with chromatic number at least N(k, l) contains either Kk or an induced odd cycle of length at least 5 or an induced cycle of length at least l. Given a graph with large chromatic number, it is natural to ask whether it must contain induced subgraphs with particular properties. One possibility is that the graph contains a large clique. If this is not the case, however, are there other graphs that G must then contain as induced subgraphs? For instance, given H and k, does every graph of sufficiently large chromatic number contain either Kk or an induced copy of H? Of course, there are graphs with arbitrarily large chromatic number and girth, so H must be acyclic. Gyárfás [1] and Sumner [8] independently made the beautiful (and difficult) conjecture that for every tree T and integer k there is an integer f(k, T) such that every graph G with χ(G) ≥ f(k, T) contains either Kk or an induced