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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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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 20 (9 self)
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It is proved that the choice number of every graph G embedded on a
Triangulated manifolds with few vertices: Combinatorial manifolds
, 2005
"... Let M be a simplicial manifold with n vertices. We call M centrally symmetric if it is invariant under an involution I of its vertex set which fixes no face of M. Obviously, the number of vertices of a centrally symmetric (triangulated) manifold is even, n = 2k, and, without loss of generality, we m ..."
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Cited by 18 (1 self)
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Let M be a simplicial manifold with n vertices. We call M centrally symmetric if it is invariant under an involution I of its vertex set which fixes no face of M. Obviously, the number of vertices of a centrally symmetric (triangulated) manifold is even, n = 2k, and, without loss of generality, we may assume that the involution is presented by the permutation I = (1 k+1)(2 k+2) · · ·(k 2k). The boundary complex ∂C ∆ k of the kdimensional crosspolytope C ∆ k is clearly centrally symmetric with respect to the standard antipodal action, and a subset F ⊆ {1, 2,...,2k} is a face of ∂C ∆ k if and only if it does not contain any minimal nonface {i, k + i} for 1 ≤ i ≤ k. Hence, every centrally symmetric manifold with 2k vertices appears as a subcomplex of the boundary complex of the kdimensional crosspolytope. Free Z2actions on spheres are at the heart of the BorsukUlam theorem, which has an abundance of applications in topology, combinatorics, functional analysis, and other areas of mathematics (see the surveys of Steinlein [50],
Nordhaus–Gaddumtype theorems for decompositions into many parts
 J. GRAPH THEORY
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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 8 (2 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 3 (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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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.