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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 40 (13 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
Layout of Graphs with Bounded TreeWidth
 2002, submitted. Stacks, Queues and Tracks: Layouts of Graph Subdivisions 41
, 2004
"... A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straight line grid) drawing of a gr ..."
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Cited by 26 (20 self)
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A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straight line grid) drawing of a graph represents the vertices by points in Z and the edges by noncrossing linesegments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of threedimensional drawing of a graph G is closely related to the queuenumber of G. In particular, if G is an nvertex member of a proper minorclosed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queuenumber.
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 14 (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, star and oriented colourings of graph subdivisions
 Discrete Math. Theoret. Comput. Sci
, 2005
"... Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This pa ..."
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Cited by 13 (5 self)
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Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G ′ be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G ′ correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χa(G ′), χs(G ′ ) and χ(G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The oriented chromatic number − → χ (G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that − → χ (G ′ ) = χ(G) whenever χ(G) ≥ 9.
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 connected subgrap ..."
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Cited by 5 (1 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.
Game Chromatic Number of Graphs
 Discrete Math
, 1998
"... We show that if a graph has acyclic chromatic number k, then its game chromatic number is at most k(k + 1). By applying the known upper bounds for the acyclic chromatic numbers of various classes of graphs, we obtain upper bounds for the game chromatic number of these classes of graphs. In particula ..."
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Cited by 5 (0 self)
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We show that if a graph has acyclic chromatic number k, then its game chromatic number is at most k(k + 1). By applying the known upper bounds for the acyclic chromatic numbers of various classes of graphs, we obtain upper bounds for the game chromatic number of these classes of graphs. In particular, since a planar graph has acyclic chromatic number at most 5, we conclude that the game chromatic number of a planar graph is at most 30, which improves the previous known upper bound for the game chromatic number of planar graphs. This research was supported in part by the National Science Council of the Republic of China under grant NSC872115M110004. y 1991 Mathematics Subject Classification. 05C20, 05C35, 05C15 Key words and phrases. Acyclic chromatic number, game chromatic number, planar graph, partial ktree Submitted to Discrete Mathematics 2 Dinski and Zhu 1 Introduction Let G be a finite graph and let X be a set of colors. We consider a modified graph coloring problem p...
Adjacency posets of planar graphs
 DISCRETE MATH
"... In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show t ..."
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Cited by 4 (3 self)
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In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertexface poset of such a graph.
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 3 (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...
Acyclic kstrong coloring of maps on surfaces
 Mathematical Notes
, 2000
"... Abstract—A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no twocolored cycles. Suppose each face of rank k, k ≥ 4, in a map on a surface S N is replaced by the clique having the same number of vertices. It is proved in [ ..."
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Cited by 3 (1 self)
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Abstract—A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no twocolored cycles. Suppose each face of rank k, k ≥ 4, in a map on a surface S N is replaced by the clique having the same number of vertices. It is proved in [1] that the resulting pseudograph admits an acyclic coloring with the number of colors depending linearly on N and k. In the present paper we prove a sharper estimate 55(−Nk) 4/7 for the number of colors provided that k ≥ 1and−N ≥ 5 7 k 4/3. Key words: graphs on surfaces, acyclic colorings, kstrong colorings. 1.
Graphs with maximum degree 5 are acyclically 7colorable
, 2011
"... An acyclic coloring is a proper coloring with the additional property that the union of any two color classes induces a forest. We show that every graph with maximum degree at most 5 has an acyclic 7coloring. We also show that every graph with maximum degree at most r has an acyclic (1 + ⌊ (r+1)2 4 ..."
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Cited by 2 (0 self)
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An acyclic coloring is a proper coloring with the additional property that the union of any two color classes induces a forest. We show that every graph with maximum degree at most 5 has an acyclic 7coloring. We also show that every graph with maximum degree at most r has an acyclic (1 + ⌊ (r+1)2 4 ⌋)coloring.