Independent sets in triangle-free cubic planar graphs

Heckman, Christopher; Thomas, Robin

doi:10.1016/j.jctb.2005.07.009

Cited by 38 publications

(42 citation statements)

References 15 publications

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“…However, a straightforward attempt to combine our ideas with those of Heckman and Thomas [11] fails, since they use the integrality of the independence number which permits to round up the obtained lower bounds.…”

Section: Resultsmentioning

confidence: 99%

Subcubic triangle-free graphs have fractional chromatic number at most 14/5

Dvořák

Sereni

Volec

2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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Section: Resultsmentioning

confidence: 99%

Subcubic triangle-free graphs have fractional chromatic number at most 14/5

Dvořák

Sereni

Volec

2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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“…Heckman and Thomas [12] showed that every triangle-free, cubic, planar graph has an independent set of size at least 3n/8, and this bound is tight. Again, forbidding faces of size greater than 6 gives a much better bound.…”

Section: Consequences For Max-cut and Independence Numbermentioning

confidence: 99%

Packing and covering odd cycles in cubic plane graphs with small faces

Nicodemos

Stehlı́k

2018

European Journal of Combinatorics

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Abstract. We show that any 3-connected cubic plane graph on n vertices, with all faces of size at most 6, can be made bipartite by deleting no more than (p + 3t)n/5 edges, where p and t are the numbers of pentagonal and triangular faces, respectively. In particular, any such graph can be made bipartite by deleting at most 12n/5 edges. This bound is tight, and we characterise the extremal graphs. We deduce tight lower bounds on the size of a maximum cut and a maximum independent set for this class of graphs. This extends and sharpens the results of Faria, Klein and Stehlík [SIAM J. Discrete Math. 26 (2012) 1458-1469.

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“…Staton [37] proved that every subcubic n-vertex triangle-free graph G satisfies α(G) ≥ 5n 14 . Furthermore, Heckman and Thomas [20] showed that if G is additionally planar, then α(G) ≥ 3n 8 . From the algorithmic side, the studying the complexity of maximization problems parameterized above polynomial-time computable lower bounds is an active area of research.…”

Section: Related Workmentioning

confidence: 99%

Large Independent Sets in Triangle-Free Planar Graphs

Dvořák

Mnich

2014

Algorithms - ESA 2014

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Every triangle-free planar graph on n vertices has an independent set of size at least (n + 1)/3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k ≥ 0, decides whether G has an independent set of size at least (n + k)/3, in time 2 O( √ k) n. Thus, the problem is fixed-parameter tractable when parameterized by k. Furthermore, as a corollary of the result used to prove the correctness of the algorithm, we show that there exists ε > 0 such that every planar graph of girth at least five on n vertices has an independent set of size at least n/(3 − ε).

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Independent sets in triangle-free cubic planar graphs

Abstract: We prove that every triangle-free planar graph on n vertices with maximum degree three has an independent set with size at least 3 8 n. This was suggested and later conjectured by Albertson, Bollobás, and Tucker.

Cited by 38 publications

References 15 publications

Subcubic triangle-free graphs have fractional chromatic number at most 14/5

Subcubic triangle-free graphs have fractional chromatic number at most 14/5

Packing and covering odd cycles in cubic plane graphs with small faces

Large Independent Sets in Triangle-Free Planar Graphs

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