Independent sets in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>{</mml:mo><mml:mstyle mathvariant="normal"><mml:mi>claw</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo>}</mml:mo></mml:mrow></mml:math>-free 4-regular graphs

Kang, Liying; Wang, Dingguo

doi:10.1016/j.disc.2014.05.019

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…Motivated by the work of Kang et al [6] and in order to obtain a lower bound on the matching number of subcubic triangle-free graphs, we showed that α(G) ≥ 3|V (G)|/10, for any (K 4 , claw, diamond)-free graph G. The bound is tight, with equality attained by the graph defined in Example 1. We leave as an open problem to characterize graphs for which equality holds.…”

Section: Conclusion and Open Problemsmentioning

confidence: 88%

“…Fraughnaugh Jones [4] improved Fajtlowicz's result in the case of triangle-free graphs with maximum degree 4 by showing that i(G) ≥ 4/13, for any such graph G. Locke and Lou [5] showed that if G is a connected K 4 -free 4-regular graph with n vertices, then α(G) ≥ (7n − 4)/26. Kang et al [6] studied the independence number of 4-regular (K 4 , claw)-free graphs. In particular, they proved the following: Theorem 8 (Kang et al [6]).…”

Section: Independence Numbermentioning

confidence: 99%

“…Kang et al [6] studied the independence number of 4-regular (K 4 , claw)-free graphs. In particular, they proved the following: Theorem 8 (Kang et al [6]). If G is a connected (K 4 , claw)-free 4-regular graph on n vertices then, apart from three exceptions, α(G) ≥ (8n − 3)/27.…”

Section: Independence Numbermentioning

confidence: 99%

“…Following this result, several authors considered the problem of finding tight lower bounds for the independence number of graphs having bounded maximum degree and not containing cliques on 3 or 4 vertices [1,2,3,4,5]. Kang et al [6] showed that if G is a connected (K 4 , claw)-free 4-regular graph on n vertices then, apart from three exceptions, α(G) ≥ (8n − 3)/27. Motivated by this result, we show that if G is a (K 4 , claw, diamond)-free graph on n vertices, then α(G) ≥ 3n/10.…”

Section: Introductionmentioning

confidence: 99%

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On line graphs of subcubic triangle-free graphs

Munaro

2017

Discrete Mathematics

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Line graphs constitute a rich and well-studied class of graphs. In this paper, we focus on three different topics related to line graphs of subcubic triangle-free graphs. First, we show that any such graph G has an independent set of size at least 3|V (G)|/10, the bound being sharp. As an immediate consequence, we have that any subcubic triangle-free graph G, with n i vertices of degree i, has a matching of size at least 3n 1 /20 + 3n 2 /10 + 9n 3 /20. Then we provide several approximate min-max theorems relating cycletransversals and cycle-packings of line graphs of subcubic triangle-free graphs. This enables us to prove Jones' Conjecture for claw-free graphs with maximum degree 4. Finally, we concentrate on the computational complexity of Feedback Vertex Set, Hamiltonian Cycle and Hamiltonian Path for subclasses of line graphs of subcubic triangle-free graphs.

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Section: Conclusion and Open Problemsmentioning

confidence: 88%

Section: Independence Numbermentioning

confidence: 99%

Section: Independence Numbermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On line graphs of subcubic triangle-free graphs

Munaro

2017

Discrete Mathematics

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“…The key ingredient for proving Theorem 1.3 is a structural result on {K 1,3 , K 4 }-free 4-regular graphs obtained very recently by Kang et al [10].…”

Section: The Proofmentioning

confidence: 99%

The clique-transversal number of a{K1,3,K4}-free 4-regular graph

2015

Discrete Mathematics

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a b s t r a c tA clique of a graph G is a complete subgraph maximal under inclusion and having at least two vertices. A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by τ c (G), is the cardinality of a minimum clique-transversal set in G. proved that τ c (G) = ⌈ n 3 ⌉ for any 2-connected {K 1,3 , K 4 }-free 4-regular graph of order n, and conjectured that τ c (G) ≤ 10n+3 27 for a connected {K 1,3 , K 4 }-free 4-regular graph of order n.In this paper, we give a short proof of the aforementioned theorem of Wang et al. and show that the above conjecture is true, apart from only three exceptions.

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Graphs G Where $$G-N[v]$$ is a Tree for Each Vertex v

Zhang,

2024

Graphs and Combinatorics

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Independent sets in{claw,K4}-free 4-regular graphs

Cited by 6 publications

References 13 publications

On line graphs of subcubic triangle-free graphs

On line graphs of subcubic triangle-free graphs

The clique-transversal number of a{K1,3,K4}-free 4-regular graph

Graphs G Where $$G-N[v]$$ is a Tree for Each Vertex v

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