11/30 (Finding Large Independent Sets in Connected Triangle-Free 3-Regular Graphs)

Fraughnaugh, Kathryn; Locke, Stephen C.

doi:10.1006/jctb.1995.1043

Cited by 22 publications

(29 citation statements)

References 8 publications

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“…We now present some elementary facts about difficult graphs. Part (iv) implies (v), for the following reason: Let v 1 , v 2 , v 3 , v 4 and v 5 be the five vertices of degree two in D, and suppose that a big set disjoint from {v 1 , v 2 } is desired. The graph D cannot contain the edges v 3 v 4 , v 3 v 5 , and v 4 v 5 , since it is triangle-free.…”

Section: Difficult Graphs and Link Graphsmentioning

confidence: 97%

“…The graph D cannot contain the edges v 3 v 4 , v 3 v 5 , and v 4 v 5 , since it is triangle-free. Hence, two of the vertices among v 3 , v 4 and v 5 are nonadjacent. If v 3 and v 4 are nonadjacent, then there is a big set I whose only vertices of degree two are v 3 and v 4 ; this big set does not contain v 1 or v 2 , as desired.…”

Section: Difficult Graphs and Link Graphsmentioning

confidence: 97%

“…The first is due to Fraughnaugh and Locke. In [4], they conjectured that if the assumption of planarity in Theorem 1.1 is weakened to the condition that G has no subgraph isomorphic to one of six fixed (nonplanar) graphs, then G has an independent set with size at least 3 8 |V (G)|. The second conjecture involves generalizing the chromatic number of a graph.…”

Section: Introductionmentioning

confidence: 98%

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Independent sets in triangle-free cubic planar graphs

Heckman

Thomas

2006

Journal of Combinatorial Theory, Series B

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Section: Difficult Graphs and Link Graphsmentioning

confidence: 97%

Section: Difficult Graphs and Link Graphsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

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Independent sets in triangle-free cubic planar graphs

Heckman

Thomas

2006

Journal of Combinatorial Theory, Series B

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“…also [7]). (Note that there are exactly two connected graphs for which this bound is best-possible [2,3,5,8] and that Fraughnaugh and Locke [4] proved that every cubic triangle-free graph G has an independent set of cardinality at least 11 30 n(G)− 2 15 , which implies that, asymptotically, 5 14 is not the correct fraction.) In order to formulate the result of Heckman and Thomas and our extension of it we need some definitions.…”

mentioning

confidence: 96%

The independence number in graphs of maximum degree three

Harant

Henning

Rautenbach

et al. 2008

Discrete Mathematics

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We prove that a K 4 -free graph G of order n, size m and maximum degree at most three has an independent set of cardinality at least 1 7 (4n − m − λ − tr ), where λ counts the number of components of G whose blocks are each either isomorphic to one of four specific graphs or edges between two of these four specific graphs and tr is the maximum number of vertex-disjoint triangles in G. Our result generalizes a bound due to Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233-237].We consider finite simple and undirected graphs G = (V, E) of order n(G) = |V | and size m(G) = |E|. The independence number α(G) of G is defined as the maximum cardinality of a set of pairwise non-adjacent vertices which is called an independent set.Our aim in the present note is to extend a result of Heckman and Thomas [6] (cf. Theorem 1) about the independence number of triangle-free graphs of maximum degree at most three to the case of graphs which may contain triangles. With their very insightful and elegant proof, Heckman and Thomas also provide a short proof for the result conjectured by Albertson, Bollobás and Tucker [1] and originally proved by Staton [9] that every trianglefree graph G of maximum degree at most three has an independent set of cardinality at least 5 14 n(G) (cf. also [7]). (Note that there are exactly two connected graphs for which this bound is best-possible [2,3,5,8] and that Fraughnaugh and Locke [4] proved that every cubic triangle-free graph G has an independent set of cardinality at least 11 30 n(G)− 2 15 , which implies that, asymptotically, 5 14 is not the correct fraction.) In order to formulate the result of Heckman and Thomas and our extension of it we need some definitions.

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“…There were only two triangle-free 3-regular connected graphs known to have independence ratio 5 14 , although Locke [13] provided an infinite family of triangle-free 3-regular connected graphs with independence ratio approaching 11 30 . Fraughnaugh and Locke [8] proved that =(G ) 7&(G )&15:(G)&4 for every connected triangle-free graph G with maximum degree 3 and provided a polynomial-time algorithm which produces an independent set I such that =(G) 7&(G)&15 |I| &4, thereby proving that the lower bound for the independence ratio for graphs in this class approaches 11 30 . In this paper, we show that =(G ) 9&(G)&26:(G)&4 for every connected K 4 -free graph with maximum degree 4.…”

mentioning

confidence: 97%