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Minimum degree and disjoint cycles in generalized claw-free graphs

2013
*
European journal of combinatorics (Print)
*

*In*this paper we extend results on

*disjoint*cycles

*in*

*claw*-

*free*

*graphs*satisfying certain

*minimum*

*degree*conditions to K 1,s -

*free*

*graphs*, normally called generalized

*claw*-

*free*

*graphs*. ... Also, the existence of 2-factors for K 1,s -

*free*

*graphs*

*with*

*minimum*

*degree*conditions will be shown. ... If G is a K 1,4 -

*free*

*graph*of order n

*with*

*minimum*

*degree*δ(G) ≥ n/3 then G contains

*at*

*least*n/3 − 4

*disjoint*

*triangles*, and if the

*minimum*

*degree*δ(G) ≥ n/2 then G contains

*at*

*least*n/3 − 3

*disjoint*...

##
###
On the Erd\H{o}s-Gyárfás conjecture in claw-free graphs

2014
*
Discussiones Mathematicae Graph Theory
*

The Erdős-Gyárfás conjecture states that every

doi:10.7151/dmgt.1732
fatcat:pjx3u33upnbwjmpq5rncbgsvva
*graph**with**minimum**degree**at**least**three*has a cycle whose length is a power of 2. ...*In*this paper, we obtain some results on this question,*in*particular for cubic*claw*-*free**graphs*. ...*In*particular,*in*1995 Erdős and Gyárfás [4] asked: If G is a*graph**with**minimum**degree**at**least**three*, does G have a cycle whose length is a power of 2? ...##
###
Page 5320 of Mathematical Reviews Vol. , Issue 2000h
[page]

2000
*
Mathematical Reviews
*

*triangles*

*in*

*claw*-

*free*

*graphs*

*with*

*minimum*

*degree*

*at*

*least*

*three*. ... Our main result is as follows: For any integer k > 2, if G is a

*claw*-

*free*

*graph*of order

*at*

*least*6(k — 1) and

*with*

*minimum*

*degree*

*at*

*least*3, then G contains k

*vertex*-

*disjoint*

*triangles*unless G is of ...

##
###
Planarization and Acyclic Colorings of Subcubic Claw-Free Graphs
[chapter]

2011
*
Lecture Notes in Computer Science
*

We show that this bound is tight by proving that the problem is NP-hard for cubic line

doi:10.1007/978-3-642-25870-1_11
fatcat:ylmxr6wzg5bzpnp5bvx3vbmudm
*graphs*(and therefore,*claw*-*free**graphs*) of maximum*degree*d ≥ 4. ... Regarding acyclic colorings, we give a polynomial-time algorithm that finds an optimal acyclic*vertex*coloring of a subcubic*claw*-*free**graph*. ... Introduction A simple, finite*graph*G is said to be*claw*-*free*if no*vertex*of G has*three*pairwise nonadjacent neighbors. It is subcubic if every*vertex*of G has*degree**at*most*three*. ...##
###
On the Erdős-Gyárfás conjecture in claw-free graphs
[article]

2013
*
arXiv
*
pre-print

The Erdős-Gyárfás conjecture states that every

arXiv:1109.5398v3
fatcat:u5vut7kttbar7pakkg7zktj7bu
*graph**with**minimum**degree**at**least**three*has a cycle whose length is a power of 2. ...*In*this paper, we obtain some results on this question,*in*particular for cubic*claw*-*free**graphs*. ... We have shown that every*claw*-*free**graph**with**minimum**degree**at**least**three*has either a C 4 or an n-hole, where n ≥ 5. ...##
###
2-factors and independent sets on claw-free graphs

2012
*
Discrete Mathematics
*

*In*this paper, we show that if G is an l-connected

*claw*-

*free*

*graph*

*with*

*minimum*

*degree*

*at*

*least*

*three*and l ∈ {2, 3}, then for any maximum independent set S, there exists a 2-factor

*in*which each cycle ... contains

*at*

*least*l − 1 vertices

*in*S. ... Because L(H 0 ) is l-connected and has

*minimum*

*degree*

*at*

*least*

*three*, H 0 is essentially l-edge-connected and the

*minimum*edge-

*degree*of H 0 is

*at*

*least*

*three*. ...

##
###
Packing 3-vertex paths in claw-free graphs and related topics

2011
*
Discrete Applied Mathematics
*

A

doi:10.1016/j.dam.2010.05.001
fatcat:74tl7ztnxrdt3ccymbrplkpbay
*claw*is a*graph**with*four vertices and*three*edges incident to the same*vertex*. A*graph*is*claw*-*free*if it has no induced subgraph isomorphic to a*claw*. Our results include the following. ... Let G be a 3-connected*claw*-*free**graph*, x a*vertex**in*G, e = xy an edge*in*G, and P a 3-*vertex*path*in*G. ... of*degree**three*belongs to a unique*triangle**in*F c and every*vertex*of every*triangle**in*F c has*degree**three**in*F c ,(f 3) F c − A is a*claw*-*free*frame of G − A (and so G − A is 2-connected and*claw*-*free*...##
###
Squared chromatic number without claws or large cliques

2018
*
Canadian mathematical bulletin
*

Let G be a

doi:10.4153/cmb-2018-024-5
fatcat:swo5wv7e3rg2lmqmbgzi7onsae
*claw*-*free**graph*on n vertices*with*clique number ω, and consider the chromatic number χ(G^2) of the square G^2 of G. ... Writing χ'_s(d) for the supremum of χ(L^2) over the line*graphs*L of simple*graphs*of maximum*degree**at*most d, we prove that χ(G^2)<χ'_s(ω) for ω∈{3,4}. ... Acknowledgments We thank Luke Postle for alerting us to a subtlety*in*an earlier version. We also thank Rémi de Joannis de Verclos and Lucas Pastor for helpful discussions*in*relation to Section 3. ...##
###
On line graphs of subcubic triangle-free graphs

2017
*
Discrete Mathematics
*

This enables us to prove Jones' Conjecture for

doi:10.1016/j.disc.2017.01.006
fatcat:kv2oaucvpvg5vjnj5t27iyntma
*claw*-*free**graphs**with*maximum*degree*4. ... 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. ...*In*this paper, we concentrate on the subclass of line*graphs*of subcubic*triangle*-*free**graphs*(a subcubic*graph*is a*graph**with*maximum*degree**at*most 3). ...##
###
Perfect Matchings in Claw-free Cubic Graphs
[article]

2009
*
arXiv
*
pre-print

We prove that every

arXiv:0906.2261v2
fatcat:ubkhda5sejdebcrlvou735wtle
*claw*-*free*cubic n-*vertex**graph**with*no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for*claw*-*free**graphs*. ... Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-*vertex**graph**with*no cutedge has*at**least*2^(cn) perfect matchings. ... Both theorems imply that every*claw*-*free*cubic*graph**with*no cutedge has*at**least*one perfect matching. ...##
###
Guarding disjoint triangles and claws in the plane

2003
*
Computational geometry
*

We consider the problem of guarding

doi:10.1016/s0925-7721(02)00130-x
fatcat:inhdug2ylrbtfmudiusyclkkpe
*triangles**in*the plane and show that (5n + 2)/4 guards can monitor the boundaries and the*free*space of n*disjoint**triangles*. ... We also consider the analogous problem for n*disjoint**claws**in*the plane and show that 3n/2 + O(1) guards are always sufficient and 3n/2 − O(1) are sometimes necessary. ... This theorem states that the maximum matching of a*graph*on n nodes covers*at**least*2 (n + 4)/3 nodes if the*graph*is simple, planar, 2connected, and the minimal*degree*is*at**least*3. Proof. ...##
###
Perfect Matchings in Claw-free Cubic Graphs

2011
*
Electronic Journal of Combinatorics
*

We prove that every

doi:10.37236/549
fatcat:iutp2oymt5d5peltzbrk7rlix4
*claw*-*free*cubic $n$-*vertex**graph**with*no cutedge has more than $2^{n/12}$ perfect matchings, thus verifying the conjecture for*claw*-*free**graphs*. ... Lovász and Plummer conjectured that there exists a fixed positive constant $c$ such that every cubic $n$-*vertex**graph**with*no cutedge has*at**least*$2^{cn}$ perfect matchings. ... Both theorems imply that every*claw*-*free*cubic*graph**with*no cutedge has*at**least*one perfect matching. ...##
###
Disjoint Paired-Dominating sets in Cubic Graphs

2019
*
Graphs and Combinatorics
*

We prove that the

doi:10.1007/s00373-019-02063-w
pmid:31631942
pmcid:PMC6777509
fatcat:mfnujdt5dvcadhcysxyxp4tbam
*vertex*set of every*claw*-*free*cubic*graph*can be partitioned into two paired-dominating sets. ... A paired-dominating set of a*graph*G is a dominating set D*with*the additional requirement that the induced subgraph G[D] contains a perfect matching. ... The*graph*invariant asked for*in*Problem 3(1) is always positive because every nonextendable matching induces a paired-dominating set whenever isolates are excluded. ...##
###
On 1-Hamilton-connected claw-free graphs

2014
*
Discrete Mathematics
*

*claw*-

*free*hourglass-

*free*

*graph*is 1-Hamilton-connected, •

*in*Section 4, we show that every 5-connected

*claw*-

*free*

*graph*

*with*

*minimum*

*degree*

*at*

*least*6 is 1-Hamilton-connected. ...

*In*the paper, we prove that (i) every 5-connected

*claw*-

*free*

*graph*

*with*

*minimum*

*degree*

*at*

*least*6 is 1-Hamilton-connected, (ii) every 4-connected

*claw*-

*free*hourglass-

*free*

*graph*is 1-Hamilton-connected. ... By Theorem J, we need to show that We are not able to extend Theorem 7 to

*claw*-

*free*

*graphs*since a closure concept that would make this possible is not known so far. ...

##
###
Total Forcing and Zero Forcing in Claw-Free Cubic Graphs
[article]

2017
*
arXiv
*
pre-print

More generally, we prove that if G is a connected,

arXiv:1708.05041v1
fatcat:hpmvrklgezeobf6ziwrhxaekii
*claw*-*free*, cubic*graph*of order n > 6, then F_t(G) <1/2n, where a*claw*-*free**graph*is a*graph*that does not contain K_1,3 as an induced subgraph. ...*At*each discrete time interval, a colored*vertex**with*exactly one non-colored neighbor forces this non-colored neighbor to be colored. ... Since there is*at*most edge of M G that joins each*vertex**in*M i+1 to S ≤i , every*vertex**in*the multigraph M i+1 has*degree**at**least*2, implying that there is a cycle*in*M i+1 . ...
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