On the tightness of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.gif" overflow="scroll"><mml:mstyle displaystyle="false"><mml:mfrac><mml:mrow><mml:mn>5</mml:mn></mml:mrow><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:mfrac></mml:mstyle></mml:math> independence ratio

Heckman, Christopher

doi:10.1016/j.disc.2007.06.044

Discrete Mathematics

2008

DOI: 10.1016/j.disc.2007.06.044

|View full text |Cite

On the tightness of the 514 independence ratio

Christopher Heckman

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2010

2018

Publication Types

Select...

Article5

Relationship

Self Cite0

Independent5

Authors

Journals

Cited by 6 publications

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Independence, odd girth, and average degree

et al. 2010

View full text Add to dashboard Cite

We prove several best-possible lower bounds in terms of the order and the average degree for the independence number of graphs which are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most 3 due to Heckman and Thomas [A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs, Discrete Math. 233 (2001), 233-237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n − m − 1)/7.

show abstract

Independence, odd girth, and average degree

et al. 2010

View full text Add to dashboard Cite

show abstract

Triangle Packings and Transversals of Some $$K_{4}$$ K 4 -Free Graphs

Munaro

2018

Graphs and Combinatorics

View full text Add to dashboard Cite

Tuza's Conjecture asserts that the minimum number τ ∆ (G) of edges of a graph G whose deletion results in a triangle-free graph is at most 2 times the maximum number ν ∆ (G) of edge-disjoint triangles of G. The complete graphs K 4 and K 5 show that the constant 2 would be best possible. Moreover, if true, the conjecture would be essentially tight even for K 4-free graphs. In this paper, we consider several subclasses of K 4-free graphs. We show that the constant 2 can be improved for them and we try to provide the optimal one. The classes we consider are of two kinds: graphs with edges in few triangles and graphs obtained by forbidding certain odd-wheels. We translate an approximate min-max relation for τ ∆ (G) and ν ∆ (G) into an equivalent one for the clique cover number and the independence number of the triangle graph of G and we provide θ-bounding functions for classes related to triangle graphs. In particular, we obtain optimal θ-bounding functions for the classes Free(K 5 , claw, diamond) and Free(P 5 , diamond, K 2,3) and a χ-bounding function for the class (banner, odd-hole, K 1,4).

show abstract

Minimum k-path vertex cover

Brešar

Kardoš

Katrenič

et al. 2011

Discrete Applied Mathematics

134

View full text Add to dashboard Cite

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

On the tightness of the 514 independence ratio

Cited by 6 publications

References 6 publications

Independence, odd girth, and average degree

Independence, odd girth, and average degree

Triangle Packings and Transversals of Some $$K_{4}$$ K 4 -Free Graphs

Minimum k-path vertex cover

Contact Info

Product

Resources

About