Results 1  10
of
58
Restricted colorings of graphs
 in Surveys in Combinatorics 1993, London Math. Soc. Lecture Notes Series 187
, 1993
"... The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, al ..."
Abstract

Cited by 76 (15 self)
 Add to MetaCart
The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems. This is mainly a survey of recent and less recent results in the area, but it contains several new results as well.
Good and Semistrong Colorings of Oriented Planar Graphs
 INF. PROCESSING LETTERS 51
, 1994
"... A kcoloring of an oriented graph G = (V, A) is an assignment c of one of the colors 1; 2; : : : ; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6= c(y). The kcoloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z). ..."
Abstract

Cited by 42 (19 self)
 Add to MetaCart
A kcoloring of an oriented graph G = (V, A) is an assignment c of one of the colors 1; 2; : : : ; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6= c(y). The kcoloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z). A kcoloring is said to be semistrong if for every vertex x of G, c(z) 6= c(t) for any pair fz; tg of vertices of N \Gamma (x). We show that every oriented planar graph has a good coloring using at most 5 \Theta 2 4 colors and that every oriented planar graph G = (V; A) with d \Gamma (x) 3 for every x 2 V has a good and semistrong coloring using at most 4 \Theta 5 \Theta 2 4 colors.
What color is your Jacobian? Graph coloring for computing derivatives
 SIAM REV
, 2005
"... Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specific ..."
Abstract

Cited by 41 (7 self)
 Add to MetaCart
Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertexcoloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrixestimation problems. The framework is based upon the viewpoint that a partition of a matrixinto structurally orthogonal groups of columns corresponds to distance2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrixas an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.
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... ..."
Abstract

Cited by 31 (19 self)
 Add to MetaCart
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...
Star Coloring of Graphs
, 2001
"... A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star ..."
Abstract

Cited by 27 (1 self)
 Add to MetaCart
A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star
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 ..."
Abstract

Cited by 14 (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 13 (5 self)
 Add to MetaCart
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 edge colorings of graphs
 Journal of Graph Theory
, 2001
"... Abstract: A proper coloring of the edges of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G, denoted by a 0 (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a 0 (G) D(G) ‡ 2 where D(G) is the maximum d ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
Abstract: A proper coloring of the edges of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G, denoted by a 0 (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a 0 (G) D(G) ‡ 2 where D(G) is the maximum degree in G. It is known that a 0 (G) 16 D(G) for any graph G. We prove that ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ
The Acyclic Edge Chromatic Number of a Random dRegular Graph is d + 1
, 2001
"... We prove the theorem from the title: the acyclic edge chromatic number of a random dregular graph is asymptotically almost surely ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
We prove the theorem from the title: the acyclic edge chromatic number of a random dregular graph is asymptotically almost surely
Colorings and Girth of Oriented Planar Graphs
 Discrete Math
, 1995
"... Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orientations of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to ..."
Abstract

Cited by 12 (7 self)
 Add to MetaCart
Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orientations of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to girth and it appears that these questions remain interesting even under large girth assumption in the range where the chromatic number is an easy invariant. In particular we prove that every orientation of any large girth planar graph is 5colorable and classify those digraphs on 3, 4 and 5 vertices which color all large girth oriented planar graphs. 1 Introduction and statement of results Given graphs G = (V; E) and G 0 = (V 0 ; E 0 ) a homomorphism from G to G 0 is any mapping f : V ! V 0 satisfying [x; y] 2 E =) [f(x); f(y)] 2 E 0 : This work has been done while the author was visiting the University of Bordeaux I and was partly supported by GA CR 2167. y With the suppo...