Results 1  10
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17
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 31 (19 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
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.
Acyclic Colorings of Products of Trees
"... We obtain bounds for the coloring numbers of products of trees for three closely related types of colorings: acyclic, distance 2, and L(2, 1). ..."
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Cited by 6 (2 self)
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We obtain bounds for the coloring numbers of products of trees for three closely related types of colorings: acyclic, distance 2, and L(2, 1).
Acyclic Colourings of 1Planar Graphs
, 1999
"... . 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. Keywords. Acyclic colouring, planar graphs, 1planar graphs. 1 ..."
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Cited by 5 (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. Keywords. Acyclic colouring, planar graphs, 1planar graphs. 1
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 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.
Colourings of the Cartesian product of graphs and multiplicative Sidon sets, preprint
"... Abstract. Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is Ffree. The Ffree chromatic number χ(G, F) of a graph G is the minimum number of colours in an Ffree colouring of G. For appropr ..."
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Cited by 2 (0 self)
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Abstract. Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is Ffree. The Ffree chromatic number χ(G, F) of a graph G is the minimum number of colours in an Ffree colouring of G. For appropriate choices of F, several wellknown types of colourings fit into this framework, including acyclic colourings, star colourings, and distance2 colourings. This paper studies Ffree colourings of the cartesian product of graphs. Let H be the cartesian product of the graphs G1, G2,..., Gd. Our main result establishes an upper bound on the Ffree chromatic number of H in terms of the maximum Ffree chromatic number of the Gi and the following numbertheoretic concept. A set S of natural numbers is kmultiplicative Sidon if ax = by implies a = b and x = y whenever x,y ∈ S and 1 ≤ a, b ≤ k. Suppose that χ(Gi, F) ≤ k and S is a kmultiplicative Sidon set of cardinality d. We prove that χ(H, F) ≤ 1+2k·max S. We then prove that the maximum density of a kmultiplicative Sidon set is Θ(1/log k). It follows that χ(H, F) ≤ O(dk log k). We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature. 1.
Colorings and Homomorphisms of Minor Closed Classes
, 2001
"... 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 is homomorphism bounded by a triangle free graph. This solves a problem posed in [14] and completes results of [13] where ..."
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Cited by 2 (0 self)
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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 is homomorphism bounded by a triangle free graph. This solves a problem posed in [14] and completes results of [13] where a similar result has been proved for Kk free graphs for k = 4; 5. It also