Results 1 - 10
of
17
Star coloring of graphs,
- J. Graph Theory
, 2004
"... a b s t r a c t For graphs of bounded maximum degree, we consider acyclic t-improper colourings, that is, colourings in which each bipartite subgraph consisting of the edges between two colour classes is acyclic, and each colour class induces a graph with maximum degree at most t. We consider the s ..."
Abstract
-
Cited by 41 (1 self)
- Add to MetaCart
a b s t r a c t For graphs of bounded maximum degree, we consider acyclic t-improper colourings, that is, colourings in which each bipartite subgraph consisting of the edges between two colour classes is acyclic, and each colour class induces a graph with maximum degree at most t. We consider the supremum, over all graphs of maximum degree at most
Coloring with no 2-colored 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 ..."
Abstract
-
Cited by 20 (0 self)
- Add to MetaCart
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 ..."
Abstract
-
Cited by 15 (6 self)
- Add to MetaCart
(Show Context)
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.
Colourings of the Cartesian product of graphs and multiplicative Sidon sets
, 2005
"... 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 F-free. The F-free chromatic number χ(G, F) of a graph G is the minimum number of colours in an F-free colouring of G. For appropriate choic ..."
Abstract
-
Cited by 8 (3 self)
- Add to MetaCart
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 F-free. The F-free chromatic number χ(G, F) of a graph G is the minimum number of colours in an F-free colouring of G. For appropriate choices of F, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies F-free 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 F-free chromatic number of H in terms of the maximum F-free chromatic number of the Gi and the following number-theoretic concept. A set S of natural numbers is k-multiplicative 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 k-multiplicative Sidon set of cardinality d. We prove that χ(H, F) ≤ 1+2k·max S. We then prove that the maximum density of a k-multiplicative 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.
Acyclic coloring of graphs of maximum degree five: nine colors are enough
"... An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G. In this paper, we show that an ..."
Abstract
-
Cited by 7 (0 self)
- Add to MetaCart
(Show Context)
An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G. In this paper, we show that any graph of maximum degree 5 has acyclic chromatic number at most 9, and we give a linear time algorithm that achieves this bound.
Acyclic Coloring of Graphs of Maximum Degree ∆
, 2005
"... An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G, and is denoted by a(G). We sho ..."
Abstract
-
Cited by 5 (0 self)
- Add to MetaCart
(Show Context)
An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G, and is denoted by a(G). We show that any graph of maximum degree ∆ has acyclic chromatic number at most ∆(∆−1) 2 for any ∆ ≥ 5, and we give an O(n ∆ 2) algorithm to acyclically color any graph of maximum degree ∆ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding a(G) ≤ ∆( ∆ − 1) + 2 [ACK + 04]. By a deeper study of the case ∆ = 5, we also show that any graph of maximum degree 5 can be acyclically colored with at most 9 colors, and give a linear time algorithm to achieve this bound.
Star coloring high girth planar graphs
"... A star coloring of a graph is a proper coloring such that no path on four vertices is 2-colored. We prove that every planar graph with girth at least 9 can be star colored using 5 colors, and that every planar graph with girth at least 14 can be star colored using 4 colors; the figure 4 is best poss ..."
Abstract
-
Cited by 1 (0 self)
- Add to MetaCart
(Show Context)
A star coloring of a graph is a proper coloring such that no path on four vertices is 2-colored. We prove that every planar graph with girth at least 9 can be star colored using 5 colors, and that every planar graph with girth at least 14 can be star colored using 4 colors; the figure 4 is best possible. We give an example of a girth 7 planar graph that requires 5 colors to star color.
New results in graph layout
, 2003
"... A track layout of a graph consists of a vertex colouring, an edge colouring, and a total or-der of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applica-tions to other models ..."
Abstract
-
Cited by 1 (1 self)
- Add to MetaCart
A track layout of a graph consists of a vertex colouring, an edge colouring, and a total or-der of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applica-tions to other models of graph layout. In particular, we improve on the results of Enomoto and Miyauchi [SIAM J. Discrete Math., 1999] regarding stack layouts of subdivisions, and give anal-ogous results for queue layouts. We solve open problems due to Felsner, Wismath, and Liotta
