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Minimum degree and disjoint cycles in generalized claw-free graphs

Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson
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  ... 
doi:10.1016/j.ejc.2012.12.002 fatcat:modp42lmlndindrcqm5c42wxrq

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

Khodakhast Bibak, Hossein Esfandiari, Pouria Salehi Nowbandegani, Mohammad Hassan Shirdareh Haghighi
2014 Discussiones Mathematicae Graph Theory  
The Erdős-Gyárfás conjecture states that every 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?  ... 
doi:10.7151/dmgt.1732 fatcat:pjx3u33upnbwjmpq5rncbgsvva

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]

Christine Cheng, Eric McDermid, Ichiro Suzuki
2011 Lecture Notes in Computer Science  
We show that this bound is tight by proving that the problem is NP-hard for cubic line 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.  ... 
doi:10.1007/978-3-642-25870-1_11 fatcat:ylmxr6wzg5bzpnp5bvx3vbmudm

On the Erdős-Gyárfás conjecture in claw-free graphs [article]

Pouria Salehi Nowbandegani, Hossein Esfandiari, Mohammad Hassan Shirdareh Haghighi, Khodakhast Bibak
2013 arXiv   pre-print
The Erdős-Gyárfás conjecture states that every 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.  ... 
arXiv:1109.5398v3 fatcat:u5vut7kttbar7pakkg7zktj7bu

2-factors and independent sets on claw-free graphs

Roman Kužel, Kenta Ozeki, Kiyoshi Yoshimoto
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.  ... 
doi:10.1016/j.disc.2011.08.020 fatcat:erof4jbzdfhhrloynfgbuclzva

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

Alexander Kelmans
2011 Discrete Applied Mathematics  
A 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  ... 
doi:10.1016/j.dam.2010.05.001 fatcat:74tl7ztnxrdt3ccymbrplkpbay

Squared chromatic number without claws or large cliques

Wouter Cames van Batenburg, Ross J. Kang
2018 Canadian mathematical bulletin  
Let G be a 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.  ... 
doi:10.4153/cmb-2018-024-5 fatcat:swo5wv7e3rg2lmqmbgzi7onsae

On line graphs of subcubic triangle-free graphs

Andrea Munaro
2017 Discrete Mathematics  
This enables us to prove Jones' Conjecture for 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).  ... 
doi:10.1016/j.disc.2017.01.006 fatcat:kv2oaucvpvg5vjnj5t27iyntma

Perfect Matchings in Claw-free Cubic Graphs [article]

Sang-il Oum
2009 arXiv   pre-print
We prove that every 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.  ... 
arXiv:0906.2261v2 fatcat:ubkhda5sejdebcrlvou735wtle

Guarding disjoint triangles and claws in the plane

Csaba D Tóth
2003 Computational geometry  
We consider the problem of guarding 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.  ... 
doi:10.1016/s0925-7721(02)00130-x fatcat:inhdug2ylrbtfmudiusyclkkpe

Perfect Matchings in Claw-free Cubic Graphs

Sang-il Oum
2011 Electronic Journal of Combinatorics  
We prove that every 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.  ... 
doi:10.37236/549 fatcat:iutp2oymt5d5peltzbrk7rlix4

Disjoint Paired-Dominating sets in Cubic Graphs

Gábor Bacsó, Csilla Bujtás, Casey Tompkins, Zsolt Tuza
2019 Graphs and Combinatorics  
We prove that the 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.  ... 
doi:10.1007/s00373-019-02063-w pmid:31631942 pmcid:PMC6777509 fatcat:mfnujdt5dvcadhcysxyxp4tbam

On 1-Hamilton-connected claw-free graphs

Tomáš Kaiser, Zdeněk Ryjáček, Petr Vrána
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.  ... 
doi:10.1016/j.disc.2013.12.009 fatcat:nggl7hpraff7jmvljsjdutw5hm

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

Randy Davila, Michael Henning
2017 arXiv   pre-print
More generally, we prove that if G is a connected, 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 .  ... 
arXiv:1708.05041v1 fatcat:hpmvrklgezeobf6ziwrhxaekii
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