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Oriented hypergraphs, stability numbers and chromatic numbers

1981
*
Discrete Mathematics
*

tnd elementary paths, to the strong

doi:10.1016/0012-365x(81)90011-x
fatcat:3cwoyjmn3jfc5d2vy6dze4sjxm
*and*weak*chromatic**number**and*the strong*and*we&*stability**number*of a*hypergraph*. ...*Oriented**hypergraphs*are defined, so that it is possible to genc&ze popositions characterizing the*chromatic**number**and*the*stability**number*of a graph by means of crientations i! ... The following two theorems characterize the*stability**number*a(G)*and*the*chromatic**number*r(G) by means of*orientations**and*elementary paths: Theorem 1 (Gallai, Milgram [l, p. 298] ). ...##
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Page 606 of Mathematical Reviews Vol. 50, Issue 3
[page]

1975
*
Mathematical Reviews
*

From the authors’ summary: “It is shown that an n-tree has

*chromatic**number*2*and*strong*chromatic**number*n+1. ... Author’s summary: “An explicit reduction is given here of the problem of determining the*stability**number*a(G) of a graph G to the problem of determining the*chromatic**number*x(H) of a graph H. ...##
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Page 3690 of Mathematical Reviews Vol. , Issue 82i
[page]

1982
*
Mathematical Reviews
*

Thomas Andreae (Berlin)
Miller, Heinrich 82i:05056

*Oriented**hypergraphs*,*stability**numbers**and**chromatic**numbers*. Discrete Math. 34 (1981), no. 3, 319-320. ... For a*hypergraph*H without loops*and*multiple edges, y(H)*and*x(H) are the two*chromatic**numbers*, termed strong*and*weak, respectively, a(H)*and*B(H) are the two*stability**numbers*, also termed strong*and*...##
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Page 8941 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

Summary: “The circular

*chromatic**number**and*the fractional*chromatic**number*are two generalizations of the ordinary chro- matic*number*of a graph. ... The*stability**number*a(H) of a*hypergraph*H is the cardinality of the largest set of vertices of H which does not contain an edge. A*hypergraph*is k-uniform if the sizes of ail its edges are k. ...##
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Page 1325 of Mathematical Reviews Vol. 50, Issue 5
[page]

1975
*
Mathematical Reviews
*

,

*Chromatic**number*of a*hypergraph*; Chapter 20, Balanced*hypergraphs**and*unimodular*hypergraphs*; Chapter 21, Matroids. 9640 COMBINATORICS 50 #19638-9644 There are new results in every chapter,*and*the ... ; Chapter 14, Kernels*and*Grundy functions; Chapter 15,*Chromatic**number*; Chapter 16, Perfect graphs; Part Two—*Hypergraphs*: Chapter 17,*Hypergraphs**and*their duals; Chapter 18, Transversals; Chapter 19 ...##
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Strict colorings of Steiner triple and quadruple systems: a survey

2003
*
Discrete Mathematics
*

The paper surveys problems, results

doi:10.1016/s0012-365x(02)00485-5
fatcat:cyk6ajriynhqddcmmj3je7dj44
*and*methods concerning the coloring of Steiner triple*and*quadruple systems viewed as mixed*hypergraphs*. ... In a D-*hypergraph*, the lower*chromatic**number*coincides with the (weak)*chromatic**number*[1, 7]*and*the upper*chromatic**number*trivially equals n. ... The parameters D , C*and*bi refer to these subsets*and*indicate those with maximum cardinality, called D-*stability*, C-*stability**and*bi-*stability**numbers*, respectively. ...##
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Page 4006 of Mathematical Reviews Vol. , Issue 83j
[page]

1983
*
Mathematical Reviews
*

Hajés conjectured that = is at least as large as the

*chromatic**number*. This was disproved by Catlin,*and*indeed Erdés*and*Fajtlowicz noted that almost all graphs provide a counterexample. B. ... This short note contains a proof of the following proposition: If the diameter of a graph G is two*and*the Betti*number*B(G) of G is even then there exists a 2-cell imbedding of G into an*orientable*surface ...##
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Page 3022 of Mathematical Reviews Vol. , Issue 84h
[page]

1984
*
Mathematical Reviews
*

to the

*stability**number*(i.e., maximum*number*of independent vertices). ... The author indicates evidence that this occurs because in small graphs the*chromatic**number*is driven by the largest clique while in large graphs the size of the largest clique*and*the*chromatic**number*...##
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Distance-two coloring of sparse graphs

2014
*
European journal of combinatorics (Print)
*

It is also shown that such classes are precisely the classes having bounded star

doi:10.1016/j.ejc.2013.09.002
fatcat:3ahs3vzlkzhuvm5tibf53d2uku
*chromatic**number*. ...*and*nowhere-dense classes. ... Acknowledgement The two authors are grateful to Omid Amini for his kind suggestions*and*remarks. ...##
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Page 2129 of Mathematical Reviews Vol. , Issue 84f
[page]

1984
*
Mathematical Reviews
*

Sotteau (Paris)
Lehel, J. 84f:05074
Covers in

*hypergraphs*. Combinatorica 2 (1982), no. 3, 305-309. Let a(H) be the*stability**number*of a*hypergraph*H=(V,E). ... Let G have*chromatic**number*x*and*let ¢ be the minimum*number*of vertices in any color class among all x-vertex-colorings of G. ...##
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Page 4074 of Mathematical Reviews Vol. 58, Issue 6
[page]

1979
*
Mathematical Reviews
*

If H

*and*H’ are two*hypergraphs*with vertex sets X¥*and*Y, their direct product H x H’ is defined as follows. ... Upper*and*lower bounds for the*chromatic**and*transversal*numbers*of a direct product are given. Stephane Foldes (Boston, Mass.) Bhat, Vasanti N. ...##
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Page 5313 of Mathematical Reviews Vol. , Issue 2000h
[page]

2000
*
Mathematical Reviews
*

(F-BORD-LB; Talence)
On the maximum average degree

*and*the*oriented**chromatic**number*of a graph. (English summary) Combinatorics*and**number*theory (Tiruchirappalli, 1996). ... If H is an*oriented*graph the*oriented**chromatic**number*of H is the minimum*number*of vertices in an*oriented*graph H’ such that there exists a homomorphism from H to H’. ...##
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Motivations and history of some of my conjectures

1997
*
Discrete Mathematics
*

', then the '

doi:10.1016/s0012-365x(96)00161-6
fatcat:weasdrfjlzh75fi5din2g2uyzm
*stability**number*', or also the 'independence*number*'), U(G) (the 'partition*number*', the minimum*number*of cliques needed to cover the vertex set),*and*4(G) (the zero-error capacity introduced ... In a paper of Lovasz [40] , this point of view was used to extend the concept of*chromatic**number*,*and*this family was called a 'set-system'. ... Faber*and*L. Lovasz., Open Problem, in: Berge ...##
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Page 6412 of Mathematical Reviews Vol. , Issue 91M
[page]

1991
*
Mathematical Reviews
*

The author translates graph- theoretic properties, such as diameter, girth,

*stability*,*chromatic**number*,*and*triangularity, of a group*hypergraph*into algebraic conditions expressed in terms of products ... This*hypergraph*is called a group*hypergraph*(or Cayley*hypergraph*). ...##
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Combinatorial games on a graph

1996
*
Discrete Mathematics
*

a certain configuration has won,

doi:10.1016/0012-365x(94)00081-s
fatcat:l7mthmnd2jekzepoivx6v5v3xa
*and*his opponent has lost. ... In fact, there are three types of games,*and*they all have a general formulation with a graph: this paper is a survey of the general results*and*problems related to these formulations. 0012-365X/96/$15.00 ... Let us recall that a simple graph G is called perfect if every induced subgraph GA satisfies ct(GA) = O(GA), where ~(G) denotes the*stability**number*of G (maximum*number*of independent vertices),*and*O ...
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