4. Conclusion The equivalence between stretchability and extendibility into an arrangement of pse... more 4. Conclusion The equivalence between stretchability and extendibility into an arrangement of pseudo-lines for contact systems of Jordan arcs does not extend to intersection systems. With more work, another equivalent condition may be given: A contact system is stretchable if and only if any subsystem has at least 3 extremal points on its unbounded region, unless it has at most an arc.
Abstract: The $ n $-th Fiedler value of a class of graphs $\ mathcal C $ is the maximum second ei... more Abstract: The $ n $-th Fiedler value of a class of graphs $\ mathcal C $ is the maximum second eigenvalue $\ lambda_2 (G) $ of a graph $ G\ in\ mathcal C $ with $ n $ vertices. In this note we relate this value to shallow minors and, as a corollary, we determine the right order of the $ n $-th Fiedler value for some minor closed classes of graphs, including the class of planar graphs.
Classes with bounded expansion, which generalise classes that exclude a topological minor, have r... more Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Nešetřil and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor.
Constrained orientations, that is orientations such that all the vertices have a prescribed indeg... more Constrained orientations, that is orientations such that all the vertices have a prescribed indegree, relates to one another many combinatorial and topological properties such as arboricity, connectivity and planarity. These orientations are the basic tool to solve planar augmentation problems 2]. We are concerned with two classes of planar graphs: maximal planar graphs (ie polyhedral graphs, triangulations) and maximal bipartite planar graphs (ie bipartite planar graphs with quadrilateral faces).
We study restricted homomorphism dualities in the context of classes with bounded expansion (whic... more We study restricted homomorphism dualities in the context of classes with bounded expansion (which are defined by means of the greatest reduced average densities—grads). This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the “approximate version” of the Hadwiger conjecture.
We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors an... more We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors and as a consequence we show that the set of all triangle free planar graphs is homomorphism bounded by a triangle free graph. It also improves the best known bound for the star chromatic number of planar graphs from 80 to 30. Our method generalizes to all minor closed classes and puts Hadwiger conjecture in yet another context.
Abstract A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of... more Abstract A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Ajtai and Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed classes). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs.
Classes of graphs with bounded expansion have been introduced in [J. Nešetřil, P. Ossona de Mende... more Classes of graphs with bounded expansion have been introduced in [J. Nešetřil, P. Ossona de Mendez, The grad of a graph and classes with bounded expansion, in: A. Raspaud, O. Delmas (Eds.), 7th International Colloquium on Graph Theory, in: Electronic Notes in Discrete Mathematics, vol. 22, Elsevier (2005), pp. 101–106; J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005)(submitted for publication)].
Abstract A link is developed between the orbits of a bi-generated permutation group and the compo... more Abstract A link is developed between the orbits of a bi-generated permutation group and the components of a permutation over an interval of Æ, these components corresponding to sub-intervals fixed by. Several bijections are established between combinatorial families whose equi-cardinality were considered as mysterious by the literature so far. A coding of pointed maps and hypermaps follows.
ARSTRACT. A family of Jordan arcs, such that two arcs arc nowhere tangent, defines a hypergraph w... more ARSTRACT. A family of Jordan arcs, such that two arcs arc nowhere tangent, defines a hypergraph whose vertices are the arcs and whose edges are the intersection points. We shall say that the hypergraph has a strong iotersection representation and, if each intersection point is interior to at most one arc. we shall say that the hypergraph has a strong contact representation.
Abstract Classes of graphs with bounded expansion have been introduced in [15],[12]. They general... more Abstract Classes of graphs with bounded expansion have been introduced in [15],[12]. They generalize both proper minor closed classes and classes with bounded degree. For any class with bounded expansion C and any integer p there exists a constant N (C, p) so that the vertex set of any graph G∈ C may be partitioned into at most N (C, p) parts, any i≤ p parts of them induce a subgraph of tree-width at most (i-1)[12](actually, of tree-depth [16] at most i, what is sensibly stronger).
A vertex colouring of a graph G is nonrepetitive if for any path P=(v1, v2,…, v2r) in G, the firs... more A vertex colouring of a graph G is nonrepetitive if for any path P=(v1, v2,…, v2r) in G, the first half is coloured differently from the second half. The Thue choice number of G is the least integer ℓ such that for every ℓ-list assignment L of G, there exists a nonrepetitive L-colouring of G. We prove that for any positive integer ℓ, there is a tree T with πch (T)> ℓ.
Abstract We give anO (| V (G)|)-time algorithm to assign vertical and horizontal segments to the ... more Abstract We give anO (| V (G)|)-time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graphG so that (i) no two segments have an interior point in common, and (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovič.
Topic maps have been developed in order to represent the structures of relationships between subj... more Topic maps have been developed in order to represent the structures of relationships between subjects, independently of resources documenting them, and to allow standard representation and interoperability of such structures. The ISO 13250 XTM specification [2] have provided a robust syntactic XML representation allowing processing and interchange of topic maps. But topic maps have so far suffered from a lack of formal description, or conceptual model.
Abstract We give a characterization of DFS cotree-critical graphs which is central to the linear ... more Abstract We give a characterization of DFS cotree-critical graphs which is central to the linear time Kuratowski finding algorithm implemented in PIGALE (Public Implementation of a Graph Algorithm Library and Editor [2]) by the authors, and deduce a justification of a very simple algorithm for finding a Kuratowski subdivision in a DFS cotree-critical graph.
Abstract: The arboricity of a graph G is the minimum number of colours needed to colour the edges... more Abstract: The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb_p (G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min (| C|; p+ 1) colours.
4. Conclusion The equivalence between stretchability and extendibility into an arrangement of pse... more 4. Conclusion The equivalence between stretchability and extendibility into an arrangement of pseudo-lines for contact systems of Jordan arcs does not extend to intersection systems. With more work, another equivalent condition may be given: A contact system is stretchable if and only if any subsystem has at least 3 extremal points on its unbounded region, unless it has at most an arc.
Abstract: The $ n $-th Fiedler value of a class of graphs $\ mathcal C $ is the maximum second ei... more Abstract: The $ n $-th Fiedler value of a class of graphs $\ mathcal C $ is the maximum second eigenvalue $\ lambda_2 (G) $ of a graph $ G\ in\ mathcal C $ with $ n $ vertices. In this note we relate this value to shallow minors and, as a corollary, we determine the right order of the $ n $-th Fiedler value for some minor closed classes of graphs, including the class of planar graphs.
Classes with bounded expansion, which generalise classes that exclude a topological minor, have r... more Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Nešetřil and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor.
Constrained orientations, that is orientations such that all the vertices have a prescribed indeg... more Constrained orientations, that is orientations such that all the vertices have a prescribed indegree, relates to one another many combinatorial and topological properties such as arboricity, connectivity and planarity. These orientations are the basic tool to solve planar augmentation problems 2]. We are concerned with two classes of planar graphs: maximal planar graphs (ie polyhedral graphs, triangulations) and maximal bipartite planar graphs (ie bipartite planar graphs with quadrilateral faces).
We study restricted homomorphism dualities in the context of classes with bounded expansion (whic... more We study restricted homomorphism dualities in the context of classes with bounded expansion (which are defined by means of the greatest reduced average densities—grads). This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the “approximate version” of the Hadwiger conjecture.
We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors an... more We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors and as a consequence we show that the set of all triangle free planar graphs is homomorphism bounded by a triangle free graph. It also improves the best known bound for the star chromatic number of planar graphs from 80 to 30. Our method generalizes to all minor closed classes and puts Hadwiger conjecture in yet another context.
Abstract A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of... more Abstract A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Ajtai and Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed classes). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs.
Classes of graphs with bounded expansion have been introduced in [J. Nešetřil, P. Ossona de Mende... more Classes of graphs with bounded expansion have been introduced in [J. Nešetřil, P. Ossona de Mendez, The grad of a graph and classes with bounded expansion, in: A. Raspaud, O. Delmas (Eds.), 7th International Colloquium on Graph Theory, in: Electronic Notes in Discrete Mathematics, vol. 22, Elsevier (2005), pp. 101–106; J. Nešetřil, P. Ossona de Mendez, Grad and classes with bounded expansion I. Decompositions, European Journal of Combinatorics (2005)(submitted for publication)].
Abstract A link is developed between the orbits of a bi-generated permutation group and the compo... more Abstract A link is developed between the orbits of a bi-generated permutation group and the components of a permutation over an interval of Æ, these components corresponding to sub-intervals fixed by. Several bijections are established between combinatorial families whose equi-cardinality were considered as mysterious by the literature so far. A coding of pointed maps and hypermaps follows.
ARSTRACT. A family of Jordan arcs, such that two arcs arc nowhere tangent, defines a hypergraph w... more ARSTRACT. A family of Jordan arcs, such that two arcs arc nowhere tangent, defines a hypergraph whose vertices are the arcs and whose edges are the intersection points. We shall say that the hypergraph has a strong iotersection representation and, if each intersection point is interior to at most one arc. we shall say that the hypergraph has a strong contact representation.
Abstract Classes of graphs with bounded expansion have been introduced in [15],[12]. They general... more Abstract Classes of graphs with bounded expansion have been introduced in [15],[12]. They generalize both proper minor closed classes and classes with bounded degree. For any class with bounded expansion C and any integer p there exists a constant N (C, p) so that the vertex set of any graph G∈ C may be partitioned into at most N (C, p) parts, any i≤ p parts of them induce a subgraph of tree-width at most (i-1)[12](actually, of tree-depth [16] at most i, what is sensibly stronger).
A vertex colouring of a graph G is nonrepetitive if for any path P=(v1, v2,…, v2r) in G, the firs... more A vertex colouring of a graph G is nonrepetitive if for any path P=(v1, v2,…, v2r) in G, the first half is coloured differently from the second half. The Thue choice number of G is the least integer ℓ such that for every ℓ-list assignment L of G, there exists a nonrepetitive L-colouring of G. We prove that for any positive integer ℓ, there is a tree T with πch (T)> ℓ.
Abstract We give anO (| V (G)|)-time algorithm to assign vertical and horizontal segments to the ... more Abstract We give anO (| V (G)|)-time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graphG so that (i) no two segments have an interior point in common, and (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovič.
Topic maps have been developed in order to represent the structures of relationships between subj... more Topic maps have been developed in order to represent the structures of relationships between subjects, independently of resources documenting them, and to allow standard representation and interoperability of such structures. The ISO 13250 XTM specification [2] have provided a robust syntactic XML representation allowing processing and interchange of topic maps. But topic maps have so far suffered from a lack of formal description, or conceptual model.
Abstract We give a characterization of DFS cotree-critical graphs which is central to the linear ... more Abstract We give a characterization of DFS cotree-critical graphs which is central to the linear time Kuratowski finding algorithm implemented in PIGALE (Public Implementation of a Graph Algorithm Library and Editor [2]) by the authors, and deduce a justification of a very simple algorithm for finding a Kuratowski subdivision in a DFS cotree-critical graph.
Abstract: The arboricity of a graph G is the minimum number of colours needed to colour the edges... more Abstract: The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb_p (G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min (| C|; p+ 1) colours.
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Papers by Patrice Ossona de Mendez