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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.
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 18 (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
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.
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.
A Guide to the Discharging Method
, 2013
"... We provide a “howto” guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results that have been proved by this technique, but rather to demystify the technique and facilitate its wider use. Along the way, we present some new proofs and new problems. ..."
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We provide a “howto” guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results that have been proved by this technique, but rather to demystify the technique and facilitate its wider use. Along the way, we present some new proofs and new problems.