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Acyclic and Oriented Chromatic Numbers of Graphs
 J. Graph Theory
, 1997
"... . The oriented chromatic number o ( ~ G) of an oriented graph ~ G = (V; A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H . The oriented chromatic number o (G) of an undirected graph G is the maximum of the oriented chromatic n ..."
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Cited by 49 (15 self)
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. The oriented chromatic number o ( ~ G) of an oriented graph ~ G = (V; A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H . The oriented chromatic number o (G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relations between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for o (G) in terms of a (G). An upper bound for o (G) in terms of a (G) was given by Raspaud and Sopena. We also give an upper bound for o (G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. Keywords. Oriented chromatic number, Acyclic chromatic number. 1
On the Maximum Average Degree and the Oriented Chromatic Number of a Graph
 Discrete Math
, 1995
"... The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In ..."
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Cited by 32 (15 self)
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The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) defined as the maximum average degree of the subgraphs of G. 1 Introduction and statement of results For every graph G we denote by V (G), with vG = jV (G)j, 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 On leave of absence from the Institute of Mathematics, Novosibirsk, 630090, Russia. With support from Engineering and Physical Sciences Research Council, UK, grant GR/K00561, and from the International Science Foundation, grant NQ4000. y This work was partially supported by the Network DIMANET of the European Union and by the grant 960101614 of the Russian F...
Coloring with no 2colored P4's
, 2004
"... A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that ..."
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Cited by 20 (0 self)
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A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that
Acyclic colourings of planar graphs with large girth
 J. London Math. Soc
, 1999
"... A proper vertexcolouring of a graph is acyclic if there are no 2coloured cycles. It is known that every planar graph is acyclically 5colourable, and that there are planar graphs with acyclic chromatic number χ a � 5 and girth g � 4. It is proved here that a planar graph satisfies χ ..."
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Cited by 17 (0 self)
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A proper vertexcolouring of a graph is acyclic if there are no 2coloured cycles. It is known that every planar graph is acyclically 5colourable, and that there are planar graphs with acyclic chromatic number χ a � 5 and girth g � 4. It is proved here that a planar graph satisfies χ
Acyclic List 7Coloring Of Planar Graphs
 KOSTOCHKA, ANDRÉ RASPAUD, AND ÉRIC SOPENA. Acyclic
, 2001
"... . The acyclic list chromatic number of every 1planar graph is proved to be at most 7 and is conjectured to be at most 5. Keywords. Acyclic coloring, List coloring, Acyclic list coloring. 1 ..."
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Cited by 12 (2 self)
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. The acyclic list chromatic number of every 1planar graph is proved to be at most 7 and is conjectured to be at most 5. Keywords. Acyclic coloring, List coloring, Acyclic list coloring. 1
Acyclic Colourings of 1Planar Graphs
, 2001
"... A graph is 1planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1planar graph is at most 20. ..."
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Cited by 11 (0 self)
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A graph is 1planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1planar graph is at most 20.
TPreserving Homomorphisms of Oriented Graphs
, 1996
"... A homomorphism of an oriented graph G = (V; A) to an oriented graph G = (V ; A ) is a mapping ' from V to V such that '(u)'(v) is an arc in G whenever uv is an arc in G. A homomorphism of G to G is said to be T preserving for some oriented graph T if for every co ..."
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Cited by 7 (2 self)
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A homomorphism of an oriented graph G = (V; A) to an oriented graph G = (V ; A ) is a mapping ' from V to V such that '(u)'(v) is an arc in G whenever uv is an arc in G. A homomorphism of G to G is said to be T preserving for some oriented graph T if for every connected subgraph H of G isomorphic to a subgraph of T , H is isomorphic to its homomorphic image in G . The T preserving oriented chromatic number ~ T (G) of an oriented graph G is the minimum number of vertices in an oriented graph G such that there exists a T preserving homomorphism of G to G . This paper discusses the existence of T preserving homomorphisms of oriented graphs. We observe that only families of graphs with bounded degree can have bounded T preserving oriented chromatic number when T has both indegree and outdegree at least two. We then provide some sufficient conditions for families of oriented graphs for having bounded T preserving oriented chromatic number when T is a directed path or a directed tree.
Acyclic Improper Colorings of Graphs
 J. Graph Theory
, 1997
"... In this paper, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph G is a mapping c from the set of vertices of G to a set of colors such that for every color i, the subgraph induced by the vertices with color i satisfies some property depending on i. ..."
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Cited by 7 (2 self)
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In this paper, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph G is a mapping c from the set of vertices of G to a set of colors such that for every color i, the subgraph induced by the vertices with color i satisfies some property depending on i. Such an improper coloring is acyclic if for every two distinct colors i and j, the subgraph induced by all the edges linking a icolored vertex and a jcolored vertex is acyclic. We prove that every outerplanar graph can be acyclically 2colored in such a way that every monochromatic subgraph has degree at most five and that this result is best possible. For planar graphs, we prove some negative results and state some open problems. 1 Introduction Let G be a graph. We denote by V (G) the vertex set of G and by E(G) the edge set of G. A coloring of G is a mapping c from V (G) to a finite set of colors C. The mapping c is a kcoloring of G if the set C has k elements. A coloring c is pro...
Negative results on acyclic improper colorings
, 2005
"... Raspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number k is at most k2 k−1. We prove that this bound is tight for k ≥ 3. We also show that some improper and/or acyclic colorings are NPcomplete on a class C of planar graphs. We try to get the most restr ..."
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Cited by 2 (1 self)
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Raspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number k is at most k2 k−1. We prove that this bound is tight for k ≥ 3. We also show that some improper and/or acyclic colorings are NPcomplete on a class C of planar graphs. We try to get the most restrictive conditions on the class C, such as having large girth and small maximum degree. In particular, we obtain the NPcompleteness of 3ACYCLIC COLORABILITY on bipartite planar graphs with maximum degree 4, and of 4ACYCLIC COLORABILITY on bipartite planar graphs with maximum degree 8.