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

Xuding Zhu
1996 Discrete Mathematics
In this paper we study 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

Jason I. Brown, Richard J. Nowakowski, Douglas Rall
1996 SIAM Journal on Discrete Mathematics
Let /(G) denote 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 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

Ágnes Tóth
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

Ágnes Tóth
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]

Wing-Kai Hon, Ton Kloks, Hsiang-Hsuan Liu, Sheung-Hung Poon, Yue-Li Wang
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]

Wing-Kai Hon, Ton Kloks, Ching-Hao Liu, Hsiang-Hsuan Liu, Sheung-Hung Poon, Yue-Li Wang
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

Geňa Hahn, Pavol Hell, Svatopluk Poljak
1995 European journal of combinatorics (Print)
ACKNOWLEDGEMENTS This research was partially supported by grants from 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

Gábor Simonyi
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]

Ágnes Tóth
2011 arXiv   pre-print
Brown, Nowakowski and Rall defined 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

Randall J. Elzinga, David A. Gregory
2010 The Electronic Journal of Linear Algebra
Weighted generalizations 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]

Piotr Krysta, Mathieu Mari, Nan Zhi
2020 arXiv   pre-print
We study 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

Ágnes Tóth
2014 Journal of combinatorial theory. Series B (Print)
Brown, Nowakowski and Rall defined 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  where they proved that for any independent set U of  ...
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