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On the bounds for the ultimate independence ratio of a graph

1996
*
Discrete Mathematics
*

In this paper we study

doi:10.1016/0012-365x(93)e0171-y
fatcat:g4c6cgnsybe5vfze632zfsivea
*the**ultimate**independence**ratio*I(G)*of**a**graph*G, which is defined as*the*limit*of**the*sequence*of**independence**ratios**of*powers*of*G. ... We construct*a**graph*G with*ultimate**independence**ratio*I(G) strictly between*the*previous known upper*bound*1/zz(G) and lower*bound*1/x(G). ... Hell*for*many valuable discussions and comments. ...##
###
The Ultimate Categorical Independence Ratio of a Graph

1996
*
SIAM Journal on Discrete Mathematics
*

Let /(G) denote

doi:10.1137/s0895480194276909
fatcat:5rnrzmzppvhrvbpevddixgz53a
*the**independence*number*of**a**graph*G. We introduce*A*(G) limk-. (Gk)/IV(G)I k, where*the*categorical*graph*product is used. This limit, surprisingly, lies in*the*range (0,1/2] U (1. ... We can show that this limit can take any such rational number, but is there any G*for*which*A*(G) is irrational?*A*useful technique*for**bounding**A*(G) is to consider special spanning subgraphs. ...*The*previous result, along with*the*next, is quite helpful in*bounding**the**ultimate*categorical*independence**ratio**of**a**graph*. THEOREM 2.3. If G is*a*regular*graph**of*degree r > 0 then*A*(G) <_ 1/2. ...##
###
Greedy approximations of independent sets in low degree graphs
[chapter]

1995
*
Lecture Notes in Computer Science
*

We investigate

doi:10.1007/bfb0015418
fatcat:envsczrgtvay5axvt432xd73xq
*the*power*of**a*family*of*greedy algorithms*for**the**independent*set problem in cubic*graphs*and*graphs**of*maximum degree three. ... We also show certain inherent limitations in*the*power*of*this family*of*algorithm: any algorithm that greedily selects vertices*of*minimum has*a*performance*ratio*at least 1:25*on*degree-three*graphs*, ... Acknowledgments We are much indebted to Professor Osamu Watanabe and Professor Jaikumar Radhakrishnan*for*informative comments and discussions. ...##
###
Page 1451 of Mathematical Reviews Vol. , Issue 97C
[page]

1997
*
Mathematical Reviews
*

{

*For**the*entire collection see MR 97b:00030. } 97¢:05092 05C35 0SC15 Zhu, Xuding (3-SFR; Burnaby, BC)*On**the**bounds**for**the**ultimate**independence**ratio**of**a**graph*. ... Summary: “In this paper we study*the**ultimate**independence**ratio*1(G)*of**a**graph*G, which is defined as*the*limit*of**the*sequence*of**independence**ratios**of*powers*of*G. ...##
###
On the ultimate lexicographic Hall-ratio

2009
*
Discrete Mathematics
*

*The*Hall-

*ratio*ρ(G)

*of*

*a*

*graph*G is

*the*

*ratio*

*of*

*the*number

*of*vertices and

*the*

*independence*number maximized over all subgraphs

*of*G. ... Here we prove

*the*conjecture

*of*Simonyi stating that

*the*

*ultimate*lexicographic Hall-

*ratio*equals

*the*fractional chromatic number

*for*all

*graphs*. ... Acknowledgements

*The*author is grateful to Gábor Simonyi

*for*helpful discussions throughout

*the*whole time

*of*this research. ...

##
###
The Ultimate Categorical Independence Ratio of Complete Multipartite Graphs

2010
*
SIAM Journal on Discrete Mathematics
*

*The*

*independence*

*ratio*i(G)

*of*

*a*

*graph*G is

*the*

*ratio*

*of*its

*independence*number and

*the*number

*of*vertices. ...

*The*

*ultimate*categorical

*independence*

*ratio*

*of*

*a*

*graph*G is defined as lim k→∞ i(G ×k ), where G ×k denotes

*the*kth categorical power

*of*G. ... Acknowledgments I am grateful to Gábor Simonyi

*for*helpful discussions throughout

*the*whole time

*of*this research. ...

##
###
Results on independent sets in categorical products of graphs, the ultimate categorical independence ratio and the ultimate categorical independent domination ratio
[article]

2013
*
arXiv
*
pre-print

*The*

*ultimate*categorical

*independence*

*ratio*is polynomial

*for*cographs, permutation

*graphs*, interval

*graphs*,

*graphs*

*of*

*bounded*treewidth and splitgraphs. ... When G is

*a*planar

*graph*

*of*maximal degree three then alpha(G × K_4) is NP-complete. We present

*a*PTAS

*for*

*the*

*ultimate*categorical

*independence*

*ratio*

*of*planar

*graphs*. ...

*A*

*The*

*ultimate*categorical

*independence*

*ratio*

*for*some classes

*of*

*graphs*In this section we show that

*the*tensor capacity is polynomial

*for*permutation

*graphs*, interval

*graphs*, and

*graphs*

*of*

*bounded*treewidth ...

##
###
Results on Independent Sets in Categorical Products of Graphs, the Ultimate Categorical Independence Ratio and the Ultimate Categorical Independent Domination Ratio
[chapter]

2014
*
Lecture Notes in Computer Science
*

*The*

*ultimate*categorical

*independence*

*ratio*is polynomial

*for*cographs, permutation

*graphs*, interval

*graphs*,

*graphs*

*of*

*bounded*treewidth and splitgraphs. ... When G is

*a*planar

*graph*

*of*maximal degree three then α(G × K 4 ) is NP-complete. We present

*a*PTAS

*for*

*the*

*ultimate*categorical

*independence*

*ratio*

*of*planar

*graphs*. ...

*A*

*The*

*ultimate*categorical

*independence*

*ratio*

*for*some classes

*of*

*graphs*In this section we show that

*the*tensor capacity is polynomial

*for*permutation

*graphs*, interval

*graphs*, and

*graphs*

*of*

*bounded*treewidth ...

##
###
On the ultimate independence ratio of a graph

1995
*
European journal of combinatorics (Print)
*

ACKNOWLEDGEMENTS This research was partially supported by grants from

doi:10.1016/0195-6698(95)90030-6
fatcat:kbyygpelorc4zmxeppmtyl45vm
*the*National Science and Engineering Research Council*of*Canada, and*the*Advanced Systems Institute*of*British Columbia. ... Zhu*for*their interest and helpful discussions. ...*The*study*of**the**ultimate**independence**ratio*can be viewed in*the*spirit*of*investigating*the*limiting behaviour*of**graph*parameters under*graph*products. ...##
###
Approximations of independent sets in graphs
[chapter]

1998
*
Lecture Notes in Computer Science
*

*ratio*is also

*bounded*by O(n/log 2 n). 9

*For*

*graphs*with high

*independence*number,

*the*

*ratios*are better. ... High-

*independence*

*graphs*Gaps in

*bounds*

*on*approximability are nowhere greater than in

*the*case

*of*

*independent*sets in

*graphs*with

*a*(G) = n/k,

*for*some fixed k > 2. ...

##
###
Asymptotic values of the Hall-ratio for graph powers

2006
*
Discrete Mathematics
*

*The*Hall-

*ratio*

*of*

*a*

*graph*G is

*the*

*ratio*

*of*

*the*number

*of*vertices and

*the*

*independence*number maximized over all subgraphs

*of*G. ... We investigate asymptotic values

*of*

*the*Hall-

*ratio*with respect to different

*graph*powers. ... I also thank an anonymous referee

*for*finding several inaccuracies in

*the*first version

*of*this paper. ...

##
###
Answer to a question of Alon and Lubetzky about the ultimate categorical independence ratio
[article]

2011
*
arXiv
*
pre-print

Brown, Nowakowski and Rall defined

arXiv:1112.6172v1
fatcat:jdittap3pjh6pawqf67skj2td4
*the**ultimate*categorical*independence**ratio**of**a**graph*G as*A*(G)=_k→∞ i(G^× k), where i(G)=α (G)/|V(G)| denotes*the**independence**ratio**of**a**graph*G, and G^× k is*the*... Let*a*(G)=|U|/|U|+|N_G(U)|: U is an*independent*set*of*G, where N_G(U) is*the*neighborhood*of*U in G. ...*The*kth categorical power G ×k is*the*k-fold categorical product*of*G.*The**ultimate*categorical*independence**ratio**of**a**graph*G is defined as*A*(G) = lim k→∞ i(G ×k ). ...##
###
Weighted matrix eigenvalue bounds on the independence number of a graph

2010
*
The Electronic Journal of Linear Algebra
*

Weighted generalizations

doi:10.13001/1081-3810.1388
fatcat:tzjfejwjqnghxhr3u7xxizowqa
*of*Hoffman's*ratio**bound**on**the**independence*number*of**a*regular*graph*are surveyed. Several known*bounds*are reviewed as special cases*of*modest extensions. ...*The*survey concludes with some observations*on**graphs*that attain*a*weighted version*of**a**bound**of*Cvetković. ...*The*authors are grateful to C. Tardif*for*his support and helpful conversations. ...##
###
Ultimate greedy approximation of independent sets in subcubic graphs
[article]

2020
*
arXiv
*
pre-print

We study

arXiv:2001.11997v1
fatcat:c36256alfjfkfmlinpievoysca
*the*approximability*of**the*maximum size*independent*set (MIS) problem in*bounded*degree*graphs*. This is*one**of**the*most classic and widely studied NP-hard optimization problems. ... With this new theory we obtain*the**ultimate*approximation*ratio**of*5/4*for*greedy*on**graphs*with maximum degree 3, which completely solves*the*open problem from*the*paper by Halldorsson and Yoshihara ( ... We would like to thank Magnús Halldórsson*for*his help with verifying our 39 example and*for*his explanations to*the*results about greedy algorithms. ...##
###
On the ultimate categorical independence ratio

2014
*
Journal of combinatorial theory. Series B (Print)
*

Brown, Nowakowski and Rall defined

doi:10.1016/j.jctb.2014.02.010
fatcat:z7erk3k3hfgczekeym66crtsva
*the**ultimate*categorical*independence**ratio**of**a**graph*G as*A*(G) = lim k→∞ ... Acknowledgement*The*author is grateful to Gábor Simonyi*for*helpful discussions throughout*the*whole time*of*this research. ...*The**ultimate*categorical*independence**ratio**of**a**graph*G is defined as*A*(G) = lim This parameter was introduced by Brown, Nowakowski and Rall in [2] where they proved that*for*any*independent*set U*of*...
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