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Oriented hypergraphs, stability numbers and chromatic numbers

Heinrich Müller
1981 Discrete Mathematics  
tnd elementary paths, to the strong 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] ).  ... 
doi:10.1016/0012-365x(81)90011-x fatcat:3cwoyjmn3jfc5d2vy6dze4sjxm

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.  ... 

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  ... 

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.  ... 

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  ... 

Strict colorings of Steiner triple and quadruple systems: a survey

Lorenzo Milazzo, Zsolt Tuza, Vitaly Voloshin
2003 Discrete Mathematics  
The paper surveys problems, results 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.  ... 
doi:10.1016/s0012-365x(02)00485-5 fatcat:cyk6ajriynhqddcmmj3je7dj44

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  ... 

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  ... 

Distance-two coloring of sparse graphs

Zdeněk Dvořák, Louis Esperet
2014 European journal of combinatorics (Print)  
It is also shown that such classes are precisely the classes having bounded star chromatic number.  ...  and nowhere-dense classes.  ...  Acknowledgement The two authors are grateful to Omid Amini for his kind suggestions and remarks.  ... 
doi:10.1016/j.ejc.2013.09.002 fatcat:3ahs3vzlkzhuvm5tibf53d2uku

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.  ... 

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.  ... 

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’.  ... 

Motivations and history of some of my conjectures

Claude Berge
1997 Discrete Mathematics  
', then the '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  ... 
doi:10.1016/s0012-365x(96)00161-6 fatcat:weasdrfjlzh75fi5din2g2uyzm

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).  ... 

Combinatorial games on a graph

Claude Berge
1996 Discrete Mathematics  
a certain configuration has won, 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  ... 
doi:10.1016/0012-365x(94)00081-s fatcat:l7mthmnd2jekzepoivx6v5v3xa
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