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### On the Fractional Intersection Number of a Graph

Edward R. Scheinerman, Ann N. Trenk
1999 Graphs and Combinatorics
We ®nd the fractional intersection number of almost all graphs by considering random graphs.  ...  An intersection representation of a graph G is a function f X V G 3 2 S (where S is any set) with the property that uv e EG if and only if f u f v H h. The size of the representation is jSj.  ...  Many thanks to Noga Alon and Joel Spencer for pointing us to Janson's inequality which gives inequality Ã.  ...

### Page 6048 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews
the fractional intersection number of a graph.  ...  We find the fractional intersection number of almost all graphs by considering random graphs.” Hiroshi Maehara (Okinawa) 2000i:05129 05C65 05C40 90C35 Chernyak, A. A.  ...

### Extremal results on intersection graphs of boxes in R^d [article]

A. Martínez-Pérez, L. Montejano, D. Oliveros
2015 arXiv   pre-print
The main purpose of this paper is to study extremal results on the intersection graphs of boxes in ^d.  ...  We calculate exactly the maximal number of intersecting pairs in a family of n boxes in ^d with the property that no k+1 boxes in have a point in common.  ...  The second and third author wish to acknowledge support by CONA-CyT under project 166306, and the support of PAPIIT under project IN112614 and IN101912 respectively.  ...

### A fractional Helly theorem for boxes

I. Bárány, F. Fodor, A. Martínez-Pérez, L. Montejano, D. Oliveros, A. Pór
2015 Computational geometry
There exists a real number β(α )>0 such that if there are αn 2 intersecting pairs in F, then F contains an intersecting subfamily of size β n.  ...  Let F be a family of n axis-parallel boxes in R^d and α∈ (1-1/d,1] a real number.  ...  Acknowledgements The authors wish to acknowledge the support of this research by the Hungarian-Mexican Intergovernmental S&T Cooperation Programme grant TÉT 10-1-2011-0471 and NIH B330/479/11 "Discrete  ...

### Fractional Helly theorem for Cartesian products of convex sets [article]

Debsoumya Chakraborti, Jaehoon Kim, Jinha Kim, Minki Kim, Hong Liu
2021 arXiv   pre-print
This is a special case of a more general result on intersections of d-Leray complexes. We also provide a construction showing that our result on d-Leray complexes is optimal.  ...  In particular, we prove that given α∈ (1-1/t^d,1] and a finite family ℱ of Cartesian products of convex sets ∏_i∈[t]A_i in ℝ^td with A_i⊂ℝ^d if at least α-fraction of the (d+1)-tuples in ℱ are intersecting  ...  Acknowledgment The authors thank Alan Lew for introducing the problem about the intersection of d-Leray complexes.  ...

### Relations between the Local Chromatic Number and Its Directed Version

Gábor Simonyi, Gábor Tardos, Ambrus Zsbán
2014 Journal of Graph Theory
We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph.  ...  The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph  ...  It remains to prove the first statement of part (b), namely the tightness of our upper bound on the fractional chromatic number for graphs with a non-integer fractional directed local chromatic number.  ...

### Relations between the local chromatic number and its directed version [article]

Gábor Simonyi, Gábor Tardos, Ambrus Zsbán
2013 arXiv   pre-print
We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph.  ...  The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph  ...  It remains to prove the first statement of part (b), namely the tightness of our upper bound on the fractional chromatic number for graphs with a non-integer fractional directed local chromatic number.  ...

### Coloring circle arrangements: New 4-chromatic planar graphs [article]

Man-Kwun Chiu, Stefan Felsner, Manfred Scheucher, Felix Schröder, Raphael Steiner, Birgit Vogtenhuber
2022 arXiv   pre-print
More precisely, the fractional chromatic number χ_f(𝒜) of the arrangement graph is bounded from above by χ_f(𝒜) ≤ 3+O(1/n), where n is the number of pseudocircles of 𝒜.  ...  It is shown that the arrangement graph of every arrangement 𝒜 of pairwise intersecting pseudocircles is "close" to being 3-colorable.  ...  Fractional colorings max |V | α(G) , ω(G) ≤ χ f (G) ≤ χ b (G) b ≤ χ(G). (4) The fractional chromatic number forms a natural lower bound for the chromatic number of graphs.  ...

### Uniform s-cross-intersecting families [article]

Peter Frankl, Andrey Kupavskii
2017 arXiv   pre-print
In this paper we determine the maximum of | A|+| B| for all n. This generalizes a result of Hilton and Milner, who determined the maximum of | A|+| B| for nonempty 1-cross-intersecting families.  ...  In this paper we study a question related to the classical Erdős-Ko-Rado theorem, which states that any family of k-element subsets of the set [n] = {1,...  ...  Acknowledgements We thank the anonymous referee for carefully reading the paper and giving several suggestions that helped to improve the presentation of the results.  ...

### A note on the colorful fractional Helly theorem [article]

Minki Kim
2015 arXiv   pre-print
Helly's theorem is a classical result concerning the intersection patterns of convex sets in R^d. Two important generalizations are the colorful version and the fractional version.  ...  Recently, Bárány et al. combined the two, obtaining a colorful fractional Helly theorem. In this paper, we give an improved version of their result.  ...  I also would like to thank Dr.Ilkyoo Choi for his advice to improve the readability of the paper.  ...

### Domination in Geometric Intersection Graphs [chapter]

Thomas Erlebach, Erik Jan van Leeuwen
2008 Lecture Notes in Computer Science
On the one hand, we show that for intersection graphs of arbitrary fat objects, the dominating set problem is as hard to approximate as for general graphs.  ...  We present approximation algorithms and inapproximability results that shed new light on the approximability of the dominating set problem in geometric intersection graphs.  ...  We do not know if the shifting technique can be used to give a constant approximation (or even a PTAS) for MDS on disk graphs of arbitrary ply, because (1) there is no upper bound on the number of 'large  ...

### Page 4960 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews
The fractional match- ing number of # is v*(#) := max{}o-¢y W(E): w isa fractional matching of #}.  ...  The profile of a family ¥ of subsets on an n-element set Z is the vector (fo, fi,--:, fn) € R"*', where fj is the number of i-element members of Y.  ...

### Intersecting designs from linear programming and graphs of diameter two

Zoltán Füredi
1994 Discrete Mathematics
We investigate l-designs (regular intersecting families) and graphs of diameter 2.  ...  The new results presented here are corollaries of a recent improvement about uniform hypergraphs with maximal fractional matchings. We propose several open problems.  ...  Acknowledgement The author is grateful to the organizers of the Second Japan Conference on Graph Theory and Combinatorics, 18-22 August 1990, at Hakone, Japan, where this paper was presented.  ...

### Page 6605 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews
The authors pose an un- solved problem on the partition of an infinite graph into a finite number of triangle-free graphs.  ...  This paper is a survey of con- cepts and results on the (r — 1)-intersection graph of r-uniform hypergraphs, mainly for r = 3.  ...

### Lower bounds for intersection searching and fractional cascading in higher dimension

Bernard Chazelle, Ding Liu
2001 Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01
If the query is a plane and the input is a polytope with n vertices, can one achieve Oðk þ polylogðnÞÞ time with subcubic storage?  ...  Given an n-edge convex subdivision of the plane, is it possible to report its k intersections with a query line segment in Oðk þ polylogðnÞÞ time, using subquadratic storage?  ...  Introduction Fractional cascading is a general technique for speeding up lookup queries in catalogs associated with the nodes of a graph  .  ...
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