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On the Fractional Intersection Number of a Graph

1999
*
Graphs and Combinatorics
*

We ®nd

doi:10.1007/s003730050068
fatcat:5bp3siulenhwxjbofy34emcqla
*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]

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

2015
*
Computational geometry
*

There exists

doi:10.1016/j.comgeo.2014.09.007
fatcat:2e7dtqocifchpnlrvhxnftjv4i
*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]

2021
*
arXiv
*
pre-print

This is

arXiv:2108.09962v3
fatcat:u7abz4bdifaotafmygf5wvuose
*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

2014
*
Journal of Graph Theory
*

We show

doi:10.1002/jgt.21834
fatcat:5bswe323xjaunf3vywkuyx5awm
*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]

2013
*
arXiv
*
pre-print

We show

arXiv:1305.7473v1
fatcat:tnznyfdylndw7ahbg4vbpbjbwy
*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]

2022
*
arXiv
*
pre-print

More precisely,

arXiv:2205.08181v1
fatcat:2xhz5ddauramnlgfxukgzc5uhy
*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]

2017
*
arXiv
*
pre-print

In this paper we determine

arXiv:1611.07258v2
fatcat:kyw7mt2mlvgrbiivk6l6ifi7ae
*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]

2015
*
arXiv
*
pre-print

Helly's theorem is

arXiv:1511.05290v2
fatcat:4wkbigddivfvxl25awc2nu25pu
*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]

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

1994
*
Discrete Mathematics
*

We investigate l-designs (regular

doi:10.1016/0012-365x(92)00478-a
fatcat:yew2albzkbhsfcwbjirbb2tctm
*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

2001
*
Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01
*

If

doi:10.1145/380752.380818
dblp:conf/stoc/ChazelleL01
fatcat:qr6xnf23jzfmdnglqzddx46bpa
*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*[14] . ...
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