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Peter Avery - One of the best experts on this subject based on the ideXlab platform.
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Score sequences of Oriented Graphs
Journal of Graph Theory, 1991Co-Authors: Peter AveryAbstract:We extend Landau's concept of the score structure of a tournament to that of the score sequence of an Oriented Graph, and give a condition for an arbitrary integer sequence to be a score sequence. The proof is by construction of a specific Oriented Graph Δ(S) with given score sequence S. It is shown that Δ(S) is transitive and has the minimum number of arcs among the Oriented Graphs with score sequence S.
Alexandre Pinlou - One of the best experts on this subject based on the ideXlab platform.
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An Oriented Coloring of Planar Graphs with Girth at Least Five
Discrete Mathematics, 2009Co-Authors: Alexandre PinlouAbstract:An Oriented k-coloring of an Oriented Graph G is a homomorphism from G to an Oriented Graph H of order k. We prove that every Oriented Graph with a maximum average degree less than 10/3 and girth at least 5 has an Oriented chromatic number at most 16. This implies that every Oriented planar Graph with girth at least 5 has an Oriented chromatic number at most 16, that improves the previous known bound of 19 due to Borodin et al. [O.V. Borodin, A.V. Kostochka, J. Nešetřil, A. Raspaud, É. Sopena, On the maximum average degree and the Oriented chromatic number of a Graph, Discrete Math. 206 (1999) 77-89].
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Oriented Colorings of Partial 2-trees
Information Processing Letters, 2008Co-Authors: Pascal Ochem, Alexandre PinlouAbstract:A homomorphism from an Oriented Graph G to an Oriented Graph H is an arc-preserving mapping f from V(G) to V(H), that is f(x)f(y) is an arc in H whenever xy is an arc in G. The Oriented chromatic number of G is the minimum order of an Oriented Graph H such that G has a homomorphism to H. In this paper, we determine the Oriented chromatic number of the class of partial 2-trees for every girth g >= 3. We also give an upper bound for the Oriented chromatic number of planar Graphs with girth at least 11.
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On the Oriented chromatic index of Oriented Graphs
Journal of Graph Theory, 2008Co-Authors: Pascal Ochem, Alexandre Pinlou, Eric SopenaAbstract:International audienceA homomorphism from an Oriented Graph G to an Oriented Graph H is a mapping f from the set of vertices of G to the set of vertices of H such that f(U)f(V) is an arc in H whenever uv is an arc in G. The Oriented chromatic index of an Oriented Graph G is the minimum number of vertices in an Oriented Graph H such that there exists a homomorphism from the line diGraph LD(G) of G to H. We give upper bounds for the Oriented chromatic index of Graphs with bounded acyclic chromatic number, of planar Graphs and of Graphs with bounded degree. We also consider lower and upper bounds of Oriented chromatic number in terms of Oriented chromatic index. We finally prove that the problem of deciding whether an Oriented Graph has Oriented chromatic index at most k is polynomial time solvable if k= 4
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On the Oriented Chromatic Index of Oriented Graphs
Journal of Graph Theory, 2008Co-Authors: Pascal Ochem, Alexandre Pinlou, Eric SopenaAbstract:A homomorphism from an Oriented Graph G to an Oriented Graph H is a mapping f from the set of vertices of G to the set of vertices of H such that f(U)f(V) is an arc in H whenever uv is an arc in G. The Oriented chromatic index of an Oriented Graph G is the minimum number of vertices in an Oriented Graph H such that there exists a homomorphism from the line diGraph LD(G) of G to H. We give upper bounds for the Oriented chromatic index of Graphs with bounded acyclic chromatic number, of planar Graphs and of Graphs with bounded degree. We also consider lower and upper bounds of Oriented chromatic number in terms of Oriented chromatic index. We finally prove that the problem of deciding whether an Oriented Graph has Oriented chromatic index at most k is polynomial time solvable if k= 4.
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Oriented Colorings of Partial 2-trees
2007Co-Authors: Pascal Ochem, Alexandre PinlouAbstract:A homomorphism from an Oriented Graph G to an Oriented Graph H is an arc-preserving mapping f from V(G) to V(H), that is f(x)f(y) is an arc in H whenever xy is an arc in G. The Oriented chromatic number of G is the minimum order of an Oriented Graph H such that G has a homomorphism to H. In this paper, we determine the Oriented chromatic number of the class of partial 2-trees for every girth g>= 3.
Pascal Ochem - One of the best experts on this subject based on the ideXlab platform.
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Oriented Colorings of Partial 2-trees
Information Processing Letters, 2008Co-Authors: Pascal Ochem, Alexandre PinlouAbstract:A homomorphism from an Oriented Graph G to an Oriented Graph H is an arc-preserving mapping f from V(G) to V(H), that is f(x)f(y) is an arc in H whenever xy is an arc in G. The Oriented chromatic number of G is the minimum order of an Oriented Graph H such that G has a homomorphism to H. In this paper, we determine the Oriented chromatic number of the class of partial 2-trees for every girth g >= 3. We also give an upper bound for the Oriented chromatic number of planar Graphs with girth at least 11.
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On the Oriented chromatic index of Oriented Graphs
Journal of Graph Theory, 2008Co-Authors: Pascal Ochem, Alexandre Pinlou, Eric SopenaAbstract:International audienceA homomorphism from an Oriented Graph G to an Oriented Graph H is a mapping f from the set of vertices of G to the set of vertices of H such that f(U)f(V) is an arc in H whenever uv is an arc in G. The Oriented chromatic index of an Oriented Graph G is the minimum number of vertices in an Oriented Graph H such that there exists a homomorphism from the line diGraph LD(G) of G to H. We give upper bounds for the Oriented chromatic index of Graphs with bounded acyclic chromatic number, of planar Graphs and of Graphs with bounded degree. We also consider lower and upper bounds of Oriented chromatic number in terms of Oriented chromatic index. We finally prove that the problem of deciding whether an Oriented Graph has Oriented chromatic index at most k is polynomial time solvable if k= 4
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On the Oriented Chromatic Index of Oriented Graphs
Journal of Graph Theory, 2008Co-Authors: Pascal Ochem, Alexandre Pinlou, Eric SopenaAbstract:A homomorphism from an Oriented Graph G to an Oriented Graph H is a mapping f from the set of vertices of G to the set of vertices of H such that f(U)f(V) is an arc in H whenever uv is an arc in G. The Oriented chromatic index of an Oriented Graph G is the minimum number of vertices in an Oriented Graph H such that there exists a homomorphism from the line diGraph LD(G) of G to H. We give upper bounds for the Oriented chromatic index of Graphs with bounded acyclic chromatic number, of planar Graphs and of Graphs with bounded degree. We also consider lower and upper bounds of Oriented chromatic number in terms of Oriented chromatic index. We finally prove that the problem of deciding whether an Oriented Graph has Oriented chromatic index at most k is polynomial time solvable if k= 4.
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Oriented Colorings of Partial 2-trees
2007Co-Authors: Pascal Ochem, Alexandre PinlouAbstract:A homomorphism from an Oriented Graph G to an Oriented Graph H is an arc-preserving mapping f from V(G) to V(H), that is f(x)f(y) is an arc in H whenever xy is an arc in G. The Oriented chromatic number of G is the minimum order of an Oriented Graph H such that G has a homomorphism to H. In this paper, we determine the Oriented chromatic number of the class of partial 2-trees for every girth g>= 3.
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Oriented coloring of 2-outerplanar Graphs
Information Processing Letters, 2007Co-Authors: Louis Esperet, Pascal OchemAbstract:A Graph G is 2-outerplanar if it has a planar embedding such that the subGraph obtained by removing the vertices of the outer face is outerplanar. The Oriented chromatic number of an Oriented Graph H is defined as the minimum order of an Oriented Graph H' such that H has a homomorphism to H'. In this paper, we prove that 2-outerplanar Graphs are 4-degenerate. We also show that Oriented 2-outerplanar Graphs have a homomorphism to the Paley tournament QR67, which implies that their (strong) Oriented chromatic number is at most 67.
Eric Sopena - One of the best experts on this subject based on the ideXlab platform.
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homomorphisms and colourings of Oriented Graphs
Discrete Mathematics, 2016Co-Authors: Eric SopenaAbstract:An Oriented Graph is a loopless diGraph with no opposite arcs. An Oriented k -colouring of an Oriented Graph G ? is a partition of its set of vertices into k parts in such a way that no two adjacent vertices belong to the same part, and all the arcs connecting every two parts have the same direction. Hence, such a colouring exists if and only if G ? admits a homomorphism to some Oriented Graph of order k .The Oriented chromatic number of G ? is then defined as the smallest k for which G ? admits an Oriented k -colouring or, equivalently, as the minimum order of an Oriented Graph to which G ? admits a homomorphism.In this paper, we survey the main results about Oriented colourings and propose a few open problems.
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Upper Oriented chromatic number of undirected Graphs and Oriented colorings of product Graphs
Discussiones Mathematicae Graph Theory, 2012Co-Authors: Eric SopenaAbstract:The Oriented chromatic number of an Oriented Graph $\vec G$ is the minimum order of an Oriented Graph $\vev H$ such that $\vec G$ admits a homomorphism to $\vev H$. The Oriented chromatic number of an undirected Graph $G$ is then the greatest Oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper Oriented chromatic number of an undirected Graph $G$, defined as the minimum order of an Oriented Graph $\vev U$ such that every orientation $\vec G$ of $G$ admits a homomorphism to $\vec U$. We give some properties of this parameter, derive some general upper bounds on the ordinary and upper Oriented chromatic numbers of Cartesian, strong, direct and lexicoGraphic products of Graphs, and consider the particular case of products of paths.
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On the Oriented chromatic index of Oriented Graphs
Journal of Graph Theory, 2008Co-Authors: Pascal Ochem, Alexandre Pinlou, Eric SopenaAbstract:International audienceA homomorphism from an Oriented Graph G to an Oriented Graph H is a mapping f from the set of vertices of G to the set of vertices of H such that f(U)f(V) is an arc in H whenever uv is an arc in G. The Oriented chromatic index of an Oriented Graph G is the minimum number of vertices in an Oriented Graph H such that there exists a homomorphism from the line diGraph LD(G) of G to H. We give upper bounds for the Oriented chromatic index of Graphs with bounded acyclic chromatic number, of planar Graphs and of Graphs with bounded degree. We also consider lower and upper bounds of Oriented chromatic number in terms of Oriented chromatic index. We finally prove that the problem of deciding whether an Oriented Graph has Oriented chromatic index at most k is polynomial time solvable if k= 4
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On the Oriented Chromatic Index of Oriented Graphs
Journal of Graph Theory, 2008Co-Authors: Pascal Ochem, Alexandre Pinlou, Eric SopenaAbstract:A homomorphism from an Oriented Graph G to an Oriented Graph H is a mapping f from the set of vertices of G to the set of vertices of H such that f(U)f(V) is an arc in H whenever uv is an arc in G. The Oriented chromatic index of an Oriented Graph G is the minimum number of vertices in an Oriented Graph H such that there exists a homomorphism from the line diGraph LD(G) of G to H. We give upper bounds for the Oriented chromatic index of Graphs with bounded acyclic chromatic number, of planar Graphs and of Graphs with bounded degree. We also consider lower and upper bounds of Oriented chromatic number in terms of Oriented chromatic index. We finally prove that the problem of deciding whether an Oriented Graph has Oriented chromatic index at most k is polynomial time solvable if k= 4.
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Oriented vertex and arc colorings of outerplanar Graphs
Information Processing Letters, 2006Co-Authors: Alexandre Pinlou, Eric SopenaAbstract:A homomorphism from an Oriented Graph G to an Oriented Graph H is an arc-preserving mapping from V(G) to V(H), that is (x)(y) is an arc in H whenever xy is an arc in G. The Oriented chromatic number of G is the minimum order of an Oriented Graph H such that G has a homomorphism to H. The Oriented chromatic index of G is the minimum order of an Oriented Graph H such that the line-diGraph of G has a homomorphism to H. In this paper, we determine for every the Oriented chromatic number and the Oriented chromatic index of the class of Oriented outerplanar Graphs with girth at least
N. A. Shah - One of the best experts on this subject based on the ideXlab platform.
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SCORE SEQUENCES IN Oriented GraphS
Journal of Applied Mathematics and Computing, 2007Co-Authors: Shariefuddin Pirzada, T. A. Naikoo, N. A. ShahAbstract:An Oriented Graph is a diGraph with no symmetric pairs of directed arcs and without loops. The score of a vertex vi in an Oriented Graph D is avi ( or simply ai )= n 1+ d +i d vi, where d +i and d vi are the outdegree and indegree, respectively, of vi and n is the number of vertices in D. In this paper, we give a new proof of Avery's theorem and obtain some stronger inequalities for scores in Oriented Graphs. We also characterize strongly transitive Oriented Graphs.
