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19
On the Maximum Average Degree and the Oriented Chromatic Number of a Graph
 Discrete Math
, 1995
"... The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In ..."
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Cited by 30 (15 self)
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The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) defined as the maximum average degree of the subgraphs of G. 1 Introduction and statement of results For every graph G we denote by V (G), with vG = jV (G)j, its set of vertices and by E(G), with e G = jE(G)j, its set of arcs or edges. A homomorphism from a graph G to a graph On leave of absence from the Institute of Mathematics, Novosibirsk, 630090, Russia. With support from Engineering and Physical Sciences Research Council, UK, grant GR/K00561, and from the International Science Foundation, grant NQ4000. y This work was partially supported by the Network DIMANET of the European Union and by the grant 960101614 of the Russian F...
Dirac's MapColor Theorem for Choosability
 J. GRAPH THEORY
, 1999
"... It is proved that the choice number of every graph G embedded on a ..."
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Cited by 14 (9 self)
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It is proved that the choice number of every graph G embedded on a
Nordhaus–Gaddumtype theorems for decompositions into many parts
 J. Graph Theory
"... into many parts ..."
Graph Color Extensions: When Hadwiger's Conjecture and Embeddings Help
 Electronic J. Comb
, 2002
"... Suppose G is rcolorable and P V (G) is such that the components of G[P ] are far apart. We show that any (r + s)coloring of G[P ] in which each component is scolored extends to an (r + s)coloring of G. If G does not contract to K 5 or is planar and s 2, then any (r + s 1)coloring of P in w ..."
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Cited by 5 (1 self)
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Suppose G is rcolorable and P V (G) is such that the components of G[P ] are far apart. We show that any (r + s)coloring of G[P ] in which each component is scolored extends to an (r + s)coloring of G. If G does not contract to K 5 or is planar and s 2, then any (r + s 1)coloring of P in which each component is scolored extends to an (r + s 1)coloring of G. This result uses the Four Color Theorem and its equivalence to Hadwiger's Conjecture for k = 5. For s = 2 this provides an armative answer to a question of Thomassen. Similar results hold for coloring arbitrary graphs embedded in both orientable and nonorientable surfaces.
Extremal Problems in Graph Theory
, 1977
"... We consider generalized graph coloring and several other extremal problems in graph theory. In classical coloring theory, we color the vertices (resp. edges) of a graph requiring only that no two adjacent vertices (resp. incident edges) receive the same color. Here we consider both weakenings and st ..."
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Cited by 4 (2 self)
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We consider generalized graph coloring and several other extremal problems in graph theory. In classical coloring theory, we color the vertices (resp. edges) of a graph requiring only that no two adjacent vertices (resp. incident edges) receive the same color. Here we consider both weakenings and strengthenings of those requirements. We also construct twisted hypercubes of small radius and find the domination number of the Kneser graph K(n, k) when n ≥ 3 4 k2 + k if k is even, and when n ≥ 3 4 k2 − k − 1 4 when k is odd. The path chromatic number χP (G) of a graph G is the least number of colors with which the vertices of G can be colored so that each color class induces a disjoint union of paths. We answer some questions of Weaver and West [31] by characterizing cartesian products of cycles with path chromatic number 2.
A Map Colour Theorem for the Union of Graphs
, 2003
"... In 1890 Heawood [10] established an upper bound for the chromatic number of a graph embedded on a surface of Euler genus g 1. ..."
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Cited by 2 (0 self)
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In 1890 Heawood [10] established an upper bound for the chromatic number of a graph embedded on a surface of Euler genus g 1.
VertexMinimal Simplicial Immersions of the Klein Bottle in Three Space
"... Although the Klein bottle can not be embedded in R 3, it can be immersed there, and in more than one way. Smooth examples of these immersions have been studied extensively, but little is known about their simplicial versions. The vertices of a triangulation play a crucial role in understanding immer ..."
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Cited by 1 (0 self)
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Although the Klein bottle can not be embedded in R 3, it can be immersed there, and in more than one way. Smooth examples of these immersions have been studied extensively, but little is known about their simplicial versions. The vertices of a triangulation play a crucial role in understanding immersions, so it is reasonable to ask: How few vertices are required to immerse the Klein bottle in R 3? Several examples that use only nine vertices are given in section 3, and since any triangulation of the Klein bottle must have at least eight vertices, the question becomes: Can the Klein bottle be immersed in R 3 using only eight vertices? In this paper, we show that, in fact, eight is not enough, nine are required. The proof consists of three parts: first exhibiting examples of 9vertex immersions; second determining all possible 8vertex triangulations of K 2; and third showing that none of these can be immersed in R 3. 1.
Embedding graphs in surfaces: MacLane’s theorem for higher genus
"... Given a closed surface S, we characterise the graphs embeddable in S by an algebraic condition asserting the existence of a sparse generating set for their cycle space. When S is the sphere, the condition defaults to MacLane’s planarity criterion. ..."
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Cited by 1 (1 self)
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Given a closed surface S, we characterise the graphs embeddable in S by an algebraic condition asserting the existence of a sparse generating set for their cycle space. When S is the sphere, the condition defaults to MacLane’s planarity criterion.
Nonorientable biembeddings . . .
 DISCRETE MATHEMATICS
, 2004
"... Constructions due to Ringel show that there exists a nonorientable face 2colourable triangular embedding of the complete graph on n vertices (equivalently a nonorientable biembedding of two Steiner triple systems of order n) for all n ≡ 3 (mod 6) with n ≥ 9. We prove the corresponding existence th ..."
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Constructions due to Ringel show that there exists a nonorientable face 2colourable triangular embedding of the complete graph on n vertices (equivalently a nonorientable biembedding of two Steiner triple systems of order n) for all n ≡ 3 (mod 6) with n ≥ 9. We prove the corresponding existence theorem for n ≡ 1 (mod 6) with n ≥ 13.
Nordhaus–Gaddum Type Theorems for Decompositions into Many Parts
, 2003
"... Abstract: A kdecomposition (G1,...,Gk) of a graph G is a partition of its edge set to form k spanning subgraphs G1,...,Gk. The classical theorem of Nordhaus and Gaddum bounds (G1) þ (G2) and (G1) (G2) over all 2decompositions of Kn. For a graph parameter p, let p(k;G) denote the maximum of Pk i1 p ..."
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Abstract: A kdecomposition (G1,...,Gk) of a graph G is a partition of its edge set to form k spanning subgraphs G1,...,Gk. The classical theorem of Nordhaus and Gaddum bounds (G1) þ (G2) and (G1) (G2) over all 2decompositions of Kn. For a graph parameter p, let p(k;G) denote the maximum of Pk i1 p(Gi) over all kdecompositions of the graph G. The clique number!, chromatic number, list chromatic number ‘, and Szekeres–Wilf number satisfy!(2;Kn) (2;Kn) ‘(2;Kn) (2;Kn) n þ 1. We obtain lower and upper bounds for!(k;Kn), (k;Kn), ‘(k;Kn), and (k;Kn). The last three behave differently for large k. We also obtain lower