Finding Independent Sets inK4-Free 4-Regular Connected Graphs

Locke, Stephen C.; Feng, Ling

doi:10.1006/jctb.1997.1772

Cited by 12 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Fraughnaugh and Locke [2] proved that every connected subcubic triangle-free graph G on n vertices has α(G) ≥ 11n/30 − 2/15. 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.…”

Section: Independence Numbermentioning

confidence: 96%

“…Clearly, ν c (G) ≥ ν ∆ (G) = α(T (G)). On the other hand, Staton [1] proved that α(H) ≥ 5 14 |V (H)|, for every subcubic triangle-free graph H. Therefore, combining everything, we get…”

mentioning

confidence: 93%

“…In Section 3, we consider the independence number α(G) of a graph G, the maximum size of an independent set of G. The famous Brooks' Theorem asserts that every connected graph G which is neither a complete graph nor an odd cycle must be ∆(G)-colourable, and so α(G) ≥ |V (G)|/∆(G). 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.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On line graphs of subcubic triangle-free graphs

Munaro

2017

Discrete Mathematics

View full text Add to dashboard Cite

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.

show abstract

Section: Independence Numbermentioning

confidence: 96%

mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On line graphs of subcubic triangle-free graphs

Munaro

2017

Discrete Mathematics

View full text Add to dashboard Cite

show abstract

“…When d = 0, we seek a lower bound on the size of a maximum independent set. This has been studied intensely for various graph classes [3,14,17,19,20,25], and is the extremal version of the optimization problem Maximum Independent Set. When d = n, the desired forest has unrestricted degree.…”

Section: Introductionmentioning

confidence: 99%

Maximum Induced Forests in Graphs of Bounded Treewidth

Chappell¹,

Pelsmajer

2013

Electron. J. Combin.

View full text Add to dashboard Cite

Given a nonnegative integer $d$ and a graph $G$, let $f_d(G)$ be the maximum order of an induced forest in $G$ having maximum degree at most $d$. We seek lower bounds for $f_d(G)$ based on the order and treewidth of $G$.We show that, for all $k,d\ge 2$ and $n\ge 1$, if $G$ is a graph with order $n$ and treewidth at most $k$, then $f_d(G)\ge\lceil{(2dn+2)/(kd+d+1)}\rceil$, unless $G\in\{K_{1,1,3},K_{2,3}\}$ and $k=d=2$. We give examples that show that this bound is tight to within $1$.We conjecture a bound for $d=1$: $f_1(G) \ge\lceil{2n/(k+2)}\rceil$, which would also be tight to within $1$, and we prove it for $k\le 3$. For $k\ge 4$ the conjecture remains open, and we prove a weaker bound: $f_1(G)\ge (2n+2)/(2k+3)$. We also examine the cases $d=0$ and $k=0,1$.Lastly, we consider open problems relating to $f_d$ for graphs on a given surface, rather than graphs of bounded treewidth.

show abstract

“…As in the proof of Theorem 1, it is easy to see that G is connected and θ-critical. On the other hand, Locke and Lou [12] showed that, for any connected K 4 -free graph G with ∆(G) ≤ 4, we have α(G) ≥ 7n(G)−4…”

mentioning

confidence: 99%

Bounded clique cover of some sparse graphs

Munaro

2017

Discrete Mathematics

View full text Add to dashboard Cite

We show that f (x) = 3 2 x is a θ-bounding function for the class of subcubic graphs and that it is best possible. This generalizes a result by Henning et al. [Independent sets and matchings in subcubic graphs, Discrete Mathematics 312 (11) (2012) 1900-1910], who showed that θ(G) ≤ 3 2 α(G) for any subcubic trianglefree graph G. Moreover, we provide a θ-bounding function for the class of K 4-free graphs with maximum degree at most 4. Finally, we study the problem Clique Cover for subclasses of planar graphs and graphs with bounded maximum degree: in particular, answering a question of Cerioli et al. [Partition into cliques for cubic graphs: Planar case, complexity and approximation, Discrete Applied Mathematics 156 (12) (2008) 2270-2278], we show it admits a PTAS for planar graphs.

show abstract

Finding Independent Sets inK4-Free 4-Regular Connected Graphs

Cited by 12 publications

References 10 publications

On line graphs of subcubic triangle-free graphs

On line graphs of subcubic triangle-free graphs

Maximum Induced Forests in Graphs of Bounded Treewidth

Bounded clique cover of some sparse graphs

Contact Info

Product

Resources

About