J. Nesetril, A. Raspaud, and E. Sopena. Colorings and girth of oriented planar graphs. Discrete Math., 165-166 :519--530, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the European Union and by the grant 96 01 01614 of the Russian Foundation for Fundamental Research. z to be completed H is a mapping from V (G) to V (H) which preserves the edges (or the arcs) that is xy 2 E(G) x) y) 2 E(H) Homomorphisms of graphs have been studied in the literature [4, 6, 7, 8, 9, 11, 15, 16, 17, 18] as a generalization of graph coloring. It is not difficult to observe that an undirected graph G has chromatic number k if and only if G has a homomorphism to K k but no homomorphism to K k Gamma1 (where K n denotes the complete graph on n vertices) Therefore the chromatic number of an ....

....The proof depends on the acyclic 5colorability of planar graphs proved in [2] Despite many efforts, no better upper bound is known up to now. The study of planar graphs is thus particularly challenging in this context. This bound can be significantly decreased under some large girth assumption [15] (recall that the girth g(G) of a graph G is the length of a shortest cycle in G) The links between the oriented chromatic number and other parameters of a graph (arboricity, maximum degree, acyclic chromatic number) have been studied in [11] In this paper, we study the relationship between the ....

[Article contains additional citation context not shown here]

J. Nesetril, A. Raspaud and E. Sopena, Colorings and girth of oriented planar graphs, Research Report 1084-95, Universit'e Bordeaux I (1995).

No context found.

J. Nesetril, A. Raspaud, and E. Sopena. Colorings and girth of oriented planar graphs. Discrete Math., 165-166 :519--530, 1997.

....of the European Union and by the grant 96 01 01614 of the Russian Foundation for Fundamental Research. to be completed H is a mapping from V (G) to V (H) which preserves the edges (or the arcs) that is xy 2 E(G) x) y) 2 E(H) Homomorphisms of graphs have been studied in the literature [4, 6, 7, 8, 9, 11, 15, 16, 17, 18] as a generalization of graph coloring. It is not difficult to observe that an undirected graph G has chromatic number k if and only if G has a homomorphism to K k but no homomorphism to K k Gamma1 (where K n denotes the complete graph on n vertices) Therefore the chromatic number of an ....

....The proof depends on the acyclic 5colorability of planar graphs proved in [2] Despite many efforts, no better upper bound is known up to now. The study of planar graphs is thus particularly challenging in this context. This bound can be significantly decreased under some large girth assumption [15] (recall that the girth g(G) of a graph G is the length of a shortest cycle in G) The links between the oriented chromatic number and other parameters of a graph (arboricity, maximum degree, acyclic chromatic number) have been studied in [11] In this paper, we study the relationship between the ....

[Article contains additional citation context not shown here]

J. Nesetril, A. Raspaud and E. Sopena, Colorings and girth of oriented planar graphs, Research Report 1084-95, Universit'e Bordeaux I (1995).

.... colorable and the vertices of G will be called colors. Oriented graphs are digraphs which contain no opposite arcs. In other words, an oriented graph is obtained from an undirected graph by giving to every edge an arbitrary orientation. Homomorphisms of oriented graphs have been studied in [12, 16, 18, 19]. The oriented chromatic number of an oriented graph G is defined as the minimum number of vertices in an oriented graph H such that there exists a homomorphism of G to H. In particular, we know that classes of graphs with bounded genus, bounded degree or bounded treewidth have bounded oriented ....

....graphs is thus particularly challenging in this context. Despite many efforts, no better upper bound is known up to now (we know that there exist oriented planar graphs with oriented chromatic number at least 15) This upper bound can be significantly decreased under some large girth assumptions [16]. Let P n denote the directed n path, that is the oriented graph defined by V ( P n ) fx 0 ; x 1 ; x n g and A( P n ) fx i x i 1 ; 0 i ng. It is easy to observe that if is a homomorphism of an oriented graph G to an oriented graph G then every directed 2 path uvw in G is ....

[Article contains additional citation context not shown here]

J. Nesetril, A. Raspaud and E. Sopena, Colorings and girth of oriented planar graphs, Research Report 1084-95, Universit'e Bordeaux I (1995).

....Cedex, France October 20, 1998 Dedicated to the memory of Francois Jaeger. Abstract The homomorphisms of oriented or unoriented graphs, the oriented chromatic number, the relationship between acyclic coloring number and oriented chromatic number, have been recently intensely studied in [1, 6, 7, 19, 20, 23, 27]. For the purpose of duality, we define the notions of strong oriented colouring and antisymmetric flow. An antisymmetric flow is a flow with values in an additive abelian group which uses no opposite elements of the group. We prove that the strong oriented chromatic number s (as the modulo ....

.... This central area of graph theory led to important modifications (such as Bouchet theory of bidirected flows , see e.g. 8, 17, 31] see also [4] Here we present yet another approach which is motivated by the recently defined version of oriented chromatic numbers (of directed graphs) see e.g. [19, 20, 23]. Among others, we define notions of Antisymmetric Flow (ASF) in the analogy to Jaeger B flow) 14] strong oriented chromatic number s (as the modulo version of oriented chromatic number) and strong colourable Cycle Double Cover. All these notions depend on the particular orientation of the ....

[Article contains additional citation context not shown here]

J. Nesetril, A. Raspaud and E. Sopena. Colorings and girth of oriented planar graphs. Discrete Math. 165--166 (1--3) (1997), 519--530.

No context found.

J. Nesetril, A. Raspaud and E. Sopena. Colorings and girth of oriented planar graphs. Discrete Math. 165--166 (1--3) (1997), 519--530.

....of Nathematics, Novosibirsk, Russia 630090 J. Nesetril Dept. of Appl. Math. Charles University, Prague, Czech Republic A. Raspaud z E. Sopena x LaBRI, Universite Bordeaux I, 33405 Talence Cedex, France 1 Introduction The notion of a nice graph first was implicitly used in the papers [1, 2, 4] as a useful tool for studying oriented chromatic number of graphs. Later, in [3] nice graphs were studied for their own sake, and some further generalisations were introduced. An oriented graph G is called k nice if for every two vertices u; v (allowing u = v) and for every orientation of ....

J. Nesetril, A. Raspaud and E. Sopena. Colorings and girth of oriented planar graphs. Discrete Math., 165/166(1996), 519--530. 7

....J. Graph Theory 24, n o 4 (1997) 331 340. Written for the Spring School in Borov a Lada Finsterau 1999, supported by EU Socrates grant #50334IC 1 97 1 CZ ERASMUS IP 1 and GA CR GACR 201 99 0242 1 [36] J. Nesetril: Structural combinatorics Graph homomorphisms and their use, to appear. [38] J. Nesetril, A. Raspaud and E. Sopena: Colorings and girth of oriented planar graphs, Discrete Math. 165 166 (1997) 519 530. 40] J. Nesetril, C. Tardif: Duality Theorems for Finite Structures (characterizing gaps and good characterizations) KAM DIMATIA Series 98 407 (submitted) 41] J. ....

....some oriented graph HF which is universal for F : Definition 6 (Universal graphs) An oriented graph HF is universal for a family F of oriented graphs if every graph F in F has a homomorphism to HF . The oriented chromatic number of graphs has been studied in this way in several papers (see e.g. [7, 38, 48, 49]) It appears that some circulant graphs can be proved to be universal for some families of graphs. Recall that the circulant graph G = G(n; c 1 ; c 2 ; c ) is the directed graph defined by V (G) f0; 1; n Gamma 1g and (x; y) 2 E(G) whenever y j x c i (mod n) for some i, 1 ....

[Article contains additional citation context not shown here]

J. Nesetril, A. Raspaud and E. Sopena. Colorings and girth of oriented planar graphs, Discrete Math. 165/166 (1997), 519--530.

....graph G is defined as the maximum of the oriented chromatic numbers of its orientations. If F is a family of graphs, the oriented chromatic number (F ) of F is defined as the maximum of the oriented chromatic numbers of the graphs in F . Oriented chromatic numbers of graphs have been studied in [7, 14, 17, 18, 19]. Raspaud and Sopena proved in [18] that every graph with acyclic chromatic number at most k has oriented chromatic number at most k Theta 2 k Gamma1 . A result of Borodin [6] states that every planar graph has acyclic chromatic number at most 5. We thus get that every planar graph has oriented ....

Nesetril, J., Raspaud, A., Sopena, E.: Colorings and girth of oriented planar graphs. Discrete Math., to appear.

....a proper colouring) and (ii) for every two arcs xy and zt in E( G) c(x) c(t) c(y) 6= c(z) The oriented chromatic number ( G) of an oriented graph G is then defined as the minimum k such that G has an oriented k colouring. Oriented chromatic numbers have been studied in [9, 11, 12, 13]. In [2] we have proved the following result which gives a link between acyclic improper colourings and oriented colourings: Theorem 1 Let P 1 ; P 2 ; P k be graph properties such that for every i, 1 i k, every orientation of every graph in the family F (P i ) has oriented chromatic ....

Nesetril, J., Raspaud, A., Sopena, E.: Colorings and girth of oriented planar graphs. Discrete Math., 165/166, 519--530, 1997.

....been constructed. The gap between the lower and the upper bounds for the oriented chromatic number of planar graphs is thus still large and, despite many efforts, has not been reduced up to now. However, the upper bound can be significantly lowered when considering planar graphs with large girth [6, 18] (recall that the girth of a graph G is the smallest size of a cycle in G) More precisely, we have the following: Theorem 5 (Borodin et al. 1997) Every planar graph with girth at least 14 (resp. 8,6,5) has oriented chromatic number at most 5 (resp. 7,11,19) In fact, this result follows from a ....

....of every graph in F has a homomorphism to U . For instance, the directed cycle on three vertices is universal for the family of trees. Most of the previous results concerning upper bounds on oriented chromatic numbers have been obtained by exhibiting some special universal oriented graphs [6, 18, 20]. In particular, an oriented (non planar) graph having 80 vertices which is universal for the family of planar graphs has been constructed in [20] The existence of planar oriented graphs which are universal for families of planar graphs with high girth has been discussed in [7] The following has ....

J. Nesetril, A. Raspaud and E. Sopena. Colorings and girth of oriented planar graphs, Discrete Math. 165/166 (1997), 519--530.

....its set of vertices and by E(G) with e G = jE(G)j, its set of arcs or edges. A homomorphism from a graph G to a graph H is a mapping from V (G) to V (H) which preserves the edges (or the arcs) that is xy 2 E(G) x) y) 2 E(H) Homomorphisms of graphs have been studied in the literature [5, 7, 8, 9, 10, 12, 16, 17, 18, 19] as a generalization of graph coloring. It is not difficult to observe that an undirected graph G has chromatic number k if and only if G has a homomorphism to K k but no homomorphism to K k Gamma1 (where K n denotes the complete graph on n vertices) Therefore the chromatic number of an ....

....The proof depends on the acyclic 5colorability of planar graphs proved in [2] Despite many efforts, no better upper bound is known up to now. The study of planar graphs is thus particularly challenging in this context. This bound can be significantly decreased under some large girth assumption [16] (recall that the girth g(G) of a graph G is the length of a shortest cycle in G) The links between the oriented chromatic number and other parameters of a graph (arboricity, maximum degree, acyclic chromatic number) have been studied in [12] In this paper, we study the relationship between the ....

[Article contains additional citation context not shown here]

J. Nesetril, A. Raspaud and E. Sopena, Colorings and girth of oriented planar graphs, Discret Math. 165--166 ((1--3) (1997), 519--530.

....4, 5, 6, 7, 10] as a generalization of graph colouring. We can similarly define the oriented chromatic number o(H) of an oriented graph H as the minimum number of vertices in an oriented graph H 0 such that H has a(n oriented) homomorphism to H 0 . Oriented homomorphisms have been studied in [2, 8, 9, 11, 12]. We will often say that a graph G is H colourable if G has a homomorphism to H and the vertices of H will be called colours. A difference between undirected and directed homomorphisms is that every undirected graph G with (G) k is K k colourable, while the minimum number of vertices in an ....

....P k the class of planar oriented graphs with girth at least k. In particular, P 3 is the class of all planar oriented graphs. Evidently, P 3 oe P 4 oe P 5 : which yields that any P k universal graph is also Pm universal for every m k. The following theorem is a summary of results in [2, 9, 11, 12] related to planar graphs. Theorem 0 1. There is a P 3 universal graph on 80 vertices [11] 2. there is a P 5 universal graph on 19 vertices [2] 3. there is a P 6 universal graph on 11 vertices [2] 4. there is a P 8 universal graph on 7 vertices [2] 5. there is a P 14 universal graph ....

[Article contains additional citation context not shown here]

J. Nesetril, A. Raspaud and E. Sopena, Colorings and girth of oriented planar graphs, Discrete Math. 165--166 (1997), 519--530.

....defined as the minimum order of an oriented graph H such that G Gamma H. The oriented chromatic number of a graph is then defined as the maximum oriented chromatic number of its orientations. The bounds on the oriented chromatic number of planar graphs in terms of girth have been considered in [4]. The connection with the maximum average degree parameter, defined as the maximum of the average degrees of all subgraphs, has been studied in [2] These two papers substantially used the property that every planar graph with sufficiently large girth, or every graph with sufficiently small ....

....0 nice for every k 0 k. We say that a digraph is nice if it is k nice for some k. Recall that the circulant digraph G = G(n; a 1 ; a 2 ; a ) is the digraph defined by V (G) f0; 1; n Gamma 1g and E(G) fxy : y = x a i (mod n) 1 i g. In fact, it has been proved in [4] that every circulant digraph of the form G(n; 1; 2; d) is d n Gamma1 d Gamma1 e nice. Let G be a multigraph whose edges are p coloured and let : E(G) Gamma f1; 2; pg denote the corresponding colouring function. For every pattern Q = q 0 q 1 : q k Gamma1 in f1; 2; ....

[Article contains additional citation context not shown here]

J. Nesetril, A. Raspaud and E. Sopena. Colorings and girth of oriented planar graphs. Discrete Math. 165/166 (1997), 519--530.

....is called a H coloring of G and the vertices of H are called colors. The oriented chromatic number of an oriented graph G is the minimum number of vertices in an oriented graph H such that G is H colorable. Homomorphisms and oriented chromatic numbers of oriented graphs have been studied in [1, 3, 4, 5, 6]. In particular, it was proved in [6] that every oriented graph with maximum degree 3 has oriented chromatic number at most 16. It was also conjectured that the tight value of this bound is 7. The aim of this short note is to decrease this upper bound to 11. In Section 2 we introduce the ....

Nesetril J., A. Raspaud and E. Sopena. Colorings and girth of oriented planar graphs. Discrete Math., to appear, 1997.

....U is a digraph obtained from U by giving to every edge one of its two possible orientations. A digraph is an oriented graph if it is an orientation of some undirected graph. Thus, oriented graphs are digraphs which contain no opposite arcs. Homomorphisms of oriented graphs have been studied in [12, 16, 18, 19]. The oriented chromatic number of an oriented graph G is defined as the minimum number of vertices in an oriented graph H such that there exists a homomorphism of G to H. The oriented chromatic number of an undirected graph U is then defined as the maximum of the oriented chromatic numbers of its ....

....graphs is thus particularly challenging in this context. Despite many efforts, no better upper bound is known up to now (we know that there exist oriented planar graphs with oriented chromatic number at least 15) This upper bound can be significantly decreased under some large girth assumptions [16]. We now introduce a generalization of the notion of acyclic coloring. We shall see that this new notion is of interest in the study of T preserving homomorphisms. Recall that the girth g(U) of an undirected graph U is the length of a shortest cycle in U . The girth of an oriented graph is then ....

[Article contains additional citation context not shown here]

Nesetril J., A. Raspaud and E. Sopena. Colorings and girth of oriented planar graphs. Research Report 1084-95, Univ. Bordeaux I, 1995.

....a proper colouring) and (ii) for every two arcs xy and zt in E( G) c(x) c(t) c(y) 6= c(z) The oriented chromatic number ( G) of an oriented graph G is then defined as the minimum k such that G has an oriented k colouring. Oriented chromatic numbers have been studied in [9, 11, 12, 13]. In [2] we have proved the following result which gives a link between acyclic improper colourings and oriented colourings: Theorem 1.1 (Boiron et al. 1997) Let P 1 ; P 2 ; P k be graph properties such that for every i, 1 i k, every orientation of every graph in the family F (P i ) ....

J. Nesetril, A. Raspaud and E. Sopena, Colorings and girth of oriented planar graphs. Discrete Math. 165/166 (1997), 519--530.

No context found.

J. Nesetril, A. Raspaud and E. Sopena, Colorings and girth of oriented planar graphs, Research Report 1084-95, Universit'e Bordeaux I (1995).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC