O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena, On the maximum average degree and the oriented chromatic number of a graph, Discrete Math., 206(1999), 77-90.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... there exists a proper (in usual sense) colouring f of V (H) with m colours with the additional property that 8v; w; u; z 2 V (H) vw 2 E(H) uz 2 E(H) f(v) f(z) f(w) 6= f(u) 1) Several properties of oriented chromatic number di er from those of the (ordinary) chromatic number (see e.g. [1, 3, 5]) In this note, we observe one more distinction. Deleting a vertex or an edge from a graph decreases its chromatic number by at most one. For the oriented chromatic number, this is not true. Observation 1 (i) If for some oriented graph H and some v 2 V (H) o(H v) k, then o(H) 2k 1. On the ....

O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena, On the maximum average degree and the oriented chromatic number of a graph, Discrete Math., 206(1999), 77-90.

....paper by listing some open questions related to our work. 1. What is the minimum integer function g(p) such that every planar graph with girth at least g(p) has bounded p acyclic chromatic number By Observation 3 we know that g(p) 2p 3 for every p. However, by using techniques inspired from [4], we can prove for instance that every planar graph with girth g 12 has 3 acyclic chromatic number at most 5, but we have no general result on this problem yet. 2. Our de nition of p acyclic colorings forces to consider graphs with girth at least p 1. An alternate de nition of p acyclic ....

Borodin O.V., A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Preprint, 1995.

....oriented chromatic number of its orientations. The bounds on the oriented chromatic number of planar graphs in terms of girth have been considered in [6] The connection with the maximum average degree parameter, de ned as the maximum of the average degrees of all subgraphs, has been studied in [5]. These two papers substantially used 1 This research was supported by the National Sciences and Engineering Research Council of Canada. 2 Part of this work was done while the author was visiting LaBRI. This author nished his part while visiting Nottingham University, funded by Visiting ....

O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Discrete Math. 206 (1999), 77-89.

....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 . 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. Oriented homomorphisms have been studied in [1, 7, 8, 10, 11]. A di erence 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 oriented graph H such that every oriented graph G with o(G) k is H colourable is exponential in k. This di erence justi es ....

....by 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 P 4 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 [1, 8, 10, 11] related to planar graphs. Theorem 1 1. There is a P 3 universal graph on 80 vertices ( 10] 2. there is a P 5 universal graph on 19 vertices ( 1] 3. there is a P 6 universal graph on 11 vertices ( 1] 4. there is a P 8 universal graph on 7 vertices ( 1] 5. there is a P 14 ....

[Article contains additional citation context not shown here]

O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena, On the maximum average degree and the oriented chromatic number of a graph. Preprint 96-336, KAM Series, Charles University, Prague, (1996).

....these schools took part in the meeting. The text tries to provide a study text for (undergraduate) students and it should serve as a background for the discussions and lectures at Spring School. Some additional material and some complementary information can be found in the following articles: [7] O.V. Borodin, A.V. Kostochka, J. Ne set ril, A. Raspaud and E. Sopena: On the maximum average degree and the oriented chromatic number of a graph, Discrete math. to appear. 24] A.V. Kostochka, E. Sopena and X. Zhu. Acyclic and oriented chromatic numbers of graphs, J. Graph Theory 24, n o 4 ....

....some oriented graph HF which is universal for F : De nition 22 (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 de ned by V (G) f0; 1; n 1g and (x; y) 2 E(G) whenever y x c i (mod n) for some i, 1 i . ....

[Article contains additional citation context not shown here]

O.V. Borodin, A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph, Discrete math., to appear.

....such that G is k nice) 2) nd a way to demonstrate non nilpotency of such semigroups ( nd as simple as possible characterization of non nice graphs) Keywords. Nice graphs, Nilpotent semigroup of endomorphisms. 1 Introduction The notion of a nice graph rst was implicitly used in the papers [1, 2, 5] 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 generalizations 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 ....

O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Discrete Math., 206(1999), 77-90.

....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 e orts, has not been reduced up to now. However, the upper bound can be signi cantly 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 [6] Theorem 5 (Borodin et al. 1999) 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 ....

....thus still large and, despite many e orts, has not been reduced up to now. However, the upper bound can be signi cantly 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 [6]: Theorem 5 (Borodin et al. 1999) 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 more general theorem. The maximum average degree mad(H) of a graph H is de ned as the maximum of the average ....

[Article contains additional citation context not shown here]

O.V. Borodin, A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph, Discrete Math. 206 (1999), 77-89.

....graph H on k vertices can be viewed as an oriented k colouring of G, using the vertices of H as colours. Oriented colourings have been introduced in [7] and studied by several authors (see [9] for a general overview) In particular, various problems related to planar graphs are discussed in [1] [2] and [6] The oriented chromatic number (G) of an oriented graph G is de ned as the smallest k such that G admits an oriented k colouring or, equivalently, as the smallest order of an oriented graph H such that G admits a homomorphism to H. The oriented chromatic number (F) of a family F of ....

Borodin, O.V., Kostochka, A.V., Nesetril, J., Raspaud, A., Sopena, E.: On the maximum average degree and the oriented chromatic number of a graph, Discrete Math. 206 (1999), 77-89.

....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 ....

Borodin O.V., A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Preprint, 1995.

....graph H on k vertices can be viewed as an oriented k colouring of G, using the vertices of H as colours. Oriented colourings have been introduced in [5] and studied by several authors (see [7] for a general overview) In particular, various problems related to planar graphs are discussed in [1] [2] and [4] The oriented chromatic number (G) of an oriented graph G is de ned as the smallest k such that G admits an oriented k colouring or, equivalently, as the smallest order of an oriented graph H such that G admits a homomorphism to H. The oriented chromatic number (F) of a family F of ....

Borodin, O.V., Kostochka, A.V., Nesetril, J., Raspaud, A., Sopena, E.: On the maximum average degree and the oriented chromatic number of a graph, Discrete Math. 206 (1999), 77-89.

....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 ....

....2 83 colours. However 2 83 6 5 and this is why we gave a direct proof not using embeddings of universal oriented graph with 80 vertices. However, let us observe that the graph introduced in the proof of Theorem 6 may be thought as a subgragh of an oriented Cayley graph. Similarly as in [6, 20] we can get better results for colourings of planar graphs of a given girth. They can be summarize as follows: Theorem 9 Let G be a planar graph. Then the following holds: 1. If G has girth at least 14 then s (G) 5, 2. If G has girth at least 8 then s (G) 7, 3. If G has girth at least ....

[Article contains additional citation context not shown here]

O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud, E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Discrete Math. (to appear)

....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 ....

O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Discrete Math., to appear. 6

....these schools took part in the meeting. The text tries to provide a study text for (undergraduate) students and it should serve as a background for the discussions and lectures at Spring School. Some additional material and some complementary information can be found in the following articles: [7] O.V. Borodin, A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena: On the maximum average degree and the oriented chromatic number of a graph, Discrete math. to appear. 24] A.V. Kostochka, E. Sopena and X. Zhu. Acyclic and oriented chromatic numbers of graphs, J. Graph Theory 24, n o 4 ....

....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]

O.V. Borodin, A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph, Discrete math., to appear.

....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 ....

Borodin, O.V., Kostochka, A.V., Nesetril, J., Raspaud, A., Sopena, E.: On the maximum average degree and the oriented chromatic number of a graph. Manuscript, 1996.

....by listing some open questions related to our work. 1. What is the minimum integer function g(p) such that every planar graph with girth at least g(p) has bounded p acyclic chromatic number By Observation 3 we know that g(p) 2p Gamma 3 for every p. However, by using techniques inspired from [4], we can prove for instance that every planar graph with girth g 12 has 3 acyclic chromatic number at most 5, but we have no general result on this problem yet. 2. Our de nition of p acyclic colorings forces to consider graphs with girth at least p 1. An alternate de nition of p acyclic ....

Borodin O.V., A.V. Kostochka, J. Ne#et#il, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Preprint, 1995.

....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 ....

O.V. Borodin, A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph, Discrete Math., to appear.

....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]

O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena, On the maximum average degree and the oriented chromatic number of a graph. Preprint 96-336, KAM Series, Charles University, Prague, (1996).

....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 maximum average degree, contains either a vertex with degree one or a long path whose internal vertices have degree two. Let k be any given positive ....

O. V. Borodin, A. V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Discrete Math., to appear.

....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 ....

Borodin O.V., A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Preprint, 1995.

....by listing some open questions related to our work. 1. What is the minimum integer function g(p) such that every planar graph with girth at least g(p) has bounded p acyclic chromatic number By Observation 3 we know that g(p) 2p Gamma 3 for every p. However, by using techniques inspired from [4], we can prove for instance that every planar graph with girth g 12 has 3 acyclic chromatic number at most 5, but we have no general result on this problem yet. 2. Our definition of p acyclic colorings forces to consider graphs with girth at least p 1. An alternate definition of p acyclic ....

Borodin O.V., A.V. Kostochka, J. Nesetril, A. Raspaud and E. Sopena. On the maximum average degree and the oriented chromatic number of a graph. Preprint, 1995.

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