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30
On Nice Graphs
, 2001
"... . A digraph G is knice for some positive integer k if for every two (not necessarily distinct) vertices x and y in G and every pattern of length k, given as a sequence of pluses and minuses, there exists a walk of length k linking x to y which respects this pattern (pluses corresponding to forward ..."
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Cited by 50 (9 self)
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. A digraph G is knice for some positive integer k if for every two (not necessarily distinct) vertices x and y in G and every pattern of length k, given as a sequence of pluses and minuses, there exists a walk of length k linking x to y which respects this pattern (pluses corresponding to forward edges and minuses to backward edges). A digraph is then nice if it is knice for some k. Similarly, a multigraph H , whose edges are coloured by a set of p colours, is knice if for every two (not necessarily distinct) vertices x and y in H and every pattern of length k, given as a sequence of colours, there exists a path of length k linking x to y which respects this pattern. Such a multigraph is nice if it is knice for some k. In this paper we study the structure of nice digraphs and multigraphs. Keywords. Graph homomorphisms, Oriented graphs, Edgecolored graphs, Universal graphs. 1
The Chromatic Number of Oriented Graphs
 J. Graph Theory
, 2001
"... . We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) dened as the minimum order of an oriented graph H such that G admits a homomorphism to H . We study the chromatic number of oriented ktrees and of oriented graphs with ..."
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Cited by 49 (22 self)
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. We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) dened as the minimum order of an oriented graph H such that G admits a homomorphism to H . We study the chromatic number of oriented ktrees and of oriented graphs with bounded degree. We show that there exist oriented ktrees with chromatic number at least 2 k+1 1 and that every oriented ktree has chromatic number at most (k + 1) 2 k . For 2trees and 3trees we decrease these upper bounds respectively to 7 and 16 and show that these new bounds are tight. As a particular case, we obtain that oriented outerplanar graphs have chromatic number at most 7 and that this bound is tight too. We then show that every oriented graph with maximum degree k has chromatic number at most (2k 1) 2 2k 2 . For oriented graphs with maximum degree 2 we decrease this bound to 5 and show that this new bound is tight. For oriented graphs with maximum degree 3 we decrease this bound to 16 and conjecture that there exists no such connected graph with chromatic number greater than 7. Keywords. Graph coloring, Graph homomorphism, Oriented coloring. 1
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 30 (18 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...
Oriented Graph Coloring
 DISCRETE MATH
, 2001
"... An oriented kcoloring of an oriented graph G (that is a digraph with no cycle of length 2) is a partition of its vertex set into k subsets such that (i) no two adjacent vertices belong to the same subset and (ii) all the arcs between any two subsets have the same direction. We survey the main resu ..."
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Cited by 18 (4 self)
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An oriented kcoloring of an oriented graph G (that is a digraph with no cycle of length 2) is a partition of its vertex set into k subsets such that (i) no two adjacent vertices belong to the same subset and (ii) all the arcs between any two subsets have the same direction. We survey the main results that have been obtained on oriented graph colorings.
On Universal Graphs for Planar Oriented Graphs of a Given Girth
, 1998
"... The oriented chromatic number o(H) of an oriented graph H is dened to be the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . If each graph in a class K has a homomorphism to the same H 0 , then H 0 is Kuniversal. Let P k denote the class of orientations of pla ..."
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Cited by 10 (5 self)
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The oriented chromatic number o(H) of an oriented graph H is dened to be the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . If each graph in a class K has a homomorphism to the same H 0 , then H 0 is Kuniversal. Let P k denote the class of orientations of planar graphs with girth at least k. Clearly, P 3 P 4 P 5 . . . We discuss the existence of P k universal graphs with special properties. It is known (see [10]) that there exists a P 3 universal graph on 80 vertices. We prove here that (1) there exist no planar P 4 universal graphs; (2) there exists a planar P 16 universal graph on 6 vertices; (3) for any k, there exist no planar P k universal graphs of girth at least 6; (4) for any k, there exists a P 40k universal graph of girth at least k + 1.
There Exist Oriented Planar Graphs with Oriented Chromatic Number at Least Sixteen
, 2001
"... We prove that there exist oriented planar graphs with oriented chromatic number at least 16. Using a result of Raspaud and Sopena [Good and semistrong colorings of oriented planar graphs, Inf. Processing Letters 51, 171174, 1994], this gives that the oriented chromatic number of the family of ori ..."
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Cited by 8 (2 self)
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We prove that there exist oriented planar graphs with oriented chromatic number at least 16. Using a result of Raspaud and Sopena [Good and semistrong colorings of oriented planar graphs, Inf. Processing Letters 51, 171174, 1994], this gives that the oriented chromatic number of the family of oriented planar graphs lies between 16 and 80.
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 co ..."
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Cited by 6 (2 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.
Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k
, 2010
"... A graph G is (k, 0)colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k ≥ 1, we prove that every graph with the maximum average degree smaller than 3k+4 is (k,0)colorable. k+2 In parti ..."
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Cited by 5 (3 self)
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A graph G is (k, 0)colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k ≥ 1, we prove that every graph with the maximum average degree smaller than 3k+4 is (k,0)colorable. k+2 In particular, it follows that every planar graph with girth at least 7 is (8,0)colorable. On the other hand, we construct planar graphs with girth 6 that are not (k, 0)colorable for arbitrarily large k.
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...