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A note on triangle-free and bipartite graphs

2002
*
Discrete Mathematics
*

Using

doi:10.1016/s0012-365x(02)00511-3
fatcat:dwwha7zyd5g27dwbaymgphmv4q
*a*clever inductive counting argument Erdős, Kleitman*and*Rothschild showed that almost all*triangle*-*free**graphs*are*bipartite*, i.e., the cardinality of the two*graph*classes is asymptotically equal ... In this paper, we investigate the structure of the few*triangle*-*free**graphs*which are not*bipartite*. ... ., result*on**triangle*-*free**graphs**and*study the structure of the (negligible) subclass of those*triangle*-*free**graphs*which are not*bipartite*. ...##
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Many disjoint triangles in co-triangle-free graphs
[article]

2020
*
arXiv
*
pre-print

Our results answer

arXiv:2001.00763v2
fatcat:5wm6wn7slvhvhegqv36alyonby
*a*question of Alon*and*Linial,*and*make progress*on**a*conjecture of Erdős. ... We prove that any n-vertex*graph*whose complement is*triangle*-*free*contains n^2/12-o(n^2) edge-disjoint*triangles*. This is tight for the disjoint union of two cliques of order n/2. ... Acknowledgement I would like to thank David Conlon for helpful discussions,*and*Olga Goulko for help with setting up the computer simulation. ...##
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A note on triangle-free graphs
[article]

2011
*
arXiv
*
pre-print

We show that if G is

arXiv:1101.3188v1
fatcat:5ygextdwhjfblkek4zkmnoe6v4
*a*simple*triangle*-*free**graph*with n≥ 3 vertices, without*a*perfect matching,*and*having*a*minimum degree at least n-1/2, then G is isomorphic either to C_5 or to K_n-1/2,n+1/2. ... Clearly, every*bipartite**graph*is*a**triangle*-*free**graph*. ...*On*the other hand, in 1974, Andrásfai, Erdős*and*Sós [1] found the minimum degree condition which forces*a**triangle*-*free**graph*to be*bipartite*. ...##
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Triangle-Free Subgraphs of Random Graphs

2015
*
Electronic Notes in Discrete Mathematics
*

The Andrásfai-Erdős-Sós Theorem [2] states that all

doi:10.1016/j.endm.2015.06.055
fatcat:wvqdi4j3knd7bfyrn7tkmgafoq
*triangle*-*free**graphs**on*n vertices with minimum degree strictly greater than 2n/5 are*bipartite*. ... We prove best possible random*graph*analogues of these theorems. ... This motivates the question of which additional restrictions*on*the class of*triangle*-*free**graphs*allow for*a*bound*on*the chromatic number. ...##
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A note on bipartite subgraphs and triangle-independent sets

2017
*
Discrete Mathematics
*

Let α_1 (G) denote the maximum size of an edge set that contains at most

doi:10.1016/j.disc.2016.07.021
fatcat:ndzg3t36hncdvjhy3ju3uvoi5e
*one*edge from each*triangle*of G. Let τ_B (G) denote the minimum size of an edge set whose deletion makes G*bipartite*. ... In this*note*, we improve the bound by showing that α_1 (G) + τ_B (G) < 4403n^2/15000 for every n-vertex*graph*G. ... Lemma 8 Let G be*a*K 5 -*free**graph**on*n vertices with e edges*and*t*triangles*. ...##
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Dense induced bipartite subgraphs in triangle-free graphs
[article]

2019
*
arXiv
*
pre-print

The problem of finding dense induced

arXiv:1810.12144v3
fatcat:rlzpgnnh5nadvhokvowyl5vvbu
*bipartite*subgraphs in H-*free**graphs*has*a*long history,*and*was posed 30 years ago by Erdős, Faudree, Pach*and*Spencer. ... Complementing this result, we further obtain optimal bounds for this problem in the case of dense*triangle*-*free**graphs*,*and*we also answer*a*question of Erdős, Janson, Łuczak*and*Spencer. ... v ∈ V (G) is included in Y if*and*only if*A*v ∩ X = ∅. Since G is*triangle*-*free*, G[Y ] is*a**bipartite**graph*with*bipartition*{N (x 1 )∩Y, (N (x 2 )\N (x 1 ))∩Y }. ...##
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Euler Graphs, Triangle-Free Graphs and Bipartite Graphs in Switching Classes
[chapter]

2002
*
Lecture Notes in Computer Science
*

*A*switching class is then

*a*set of

*graphs*obtainable from

*a*given start

*graph*by applying the switching operation. ... For all three we find algorithms running in time polynomial in the number of vertices in the

*graphs*, although switching classes contain exponentially many

*graphs*. ... G τ | M (u) is

*bipartite*

*and*hence

*triangle*-

*free*. ...

##
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Bipartite induced density in triangle-free graphs
[article]

2020
*
arXiv
*
pre-print

We prove that any

arXiv:1808.02512v3
fatcat:lvzw7566tbgjnhmdt26lmubgty
*triangle*-*free**graph**on*n vertices with minimum degree at least d contains*a**bipartite*induced subgraph of minimum degree at least d^2/(2n). ... This is sharp up to*a*logarithmic factor in n. Relatedly, we show that the fractional chromatic number of any such*triangle*-*free**graph*is at most the minimum of n/d*and*(2+o(1))√(n/log n) as n→∞. ... Acknowledgements We thank Ewan Davies for his insightful remark about regular*graphs*in Conjectures 4.3*and*4.4. ...##
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The Graph Tessellation Cover Number: Extremal Bounds, Efficient Algorithms and Hardness
[chapter]

2018
*
Lecture Notes in Computer Science
*

We establish upper bounds

doi:10.1007/978-3-319-77404-6_1
fatcat:ufrymdqywze2jozmfuh6w33ifi
*on*the tessellation cover number given by the minimum between the chromatic index of the*graph**and*the chromatic number of its clique*graph**and*we show*graph*classes for which ...*On*the other hand, we improve the complexity for 2-tessellability to*a*linear-time algorithm. ...*Note*that any of its tessellations can only be formed by cliques of size two or*one*. Hence, we have an extremal result that if G is*a**triangle*-*free**graph*, then T (G) = χ (G). ...##
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Bipartite Induced Density in Triangle-Free Graphs

2020
*
Electronic Journal of Combinatorics
*

We prove that any

doi:10.37236/8650
fatcat:gkhy6dapx5c7hh4r7ql5hqekum
*triangle*-*free**graph**on*$n$ vertices with minimum degree at least $d$ contains*a**bipartite*induced subgraph of minimum degree at least $d^2/(2n)$. ... Second, any*triangle*-*free**graph**on*$n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$. ... Acknowledgements We thank Ewan Davies for his insightful remark about regular*graphs*in Conjectures 4.3*and*4.4. ...##
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Separation Choosability and Dense Bipartite Induced Subgraphs

2019
*
Combinatorics, probability & computing
*

For example, does every

doi:10.1017/s0963548319000026
fatcat:yjnunsgmvvcstlci77s3dou4hy
*triangle*-*free**graph*of minimum degree d contain*a**bipartite*induced subgraph of minimum degree Ω(log d) as d→∞? ... We show for*bipartite**graphs*that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy*and*Thron*and*, partially, of Alon. ... confirmed Conjecture 1.3)*and*have nearly settled Conjecture 1.5 (*and*thus confirmed Conjecture 1.6), in that they have established ( log d/ log log d)*bipartite*induced minimum degree in K r -*free**graphs*...##
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Triangle-free graphs and forbidden subgraphs

2002
*
Discrete Applied Mathematics
*

In particular, we introduce chordal

doi:10.1016/s0166-218x(01)00277-3
fatcat:2v7gp75pcvaozdcdmucgaiplpm
*triangle*-*free**graphs*as*a*natural superclass of chordal*bipartite**graphs**and*describe the structure of the maximal*triangle*-*free*members. ... Based*on**a*characterization statement of Pach, some results*on*the chromatic number of*triangle*-*free**graphs*with certain forbidden induced subgraphs will be reÿned by describing their structure in terms ...*A*connected*triangle*-*free*; P 5 -*free**graph*is*bipartite*; or it is maximal*triangle*-*free*;*and*homomorphic with C 5 . ...##
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On a bipartition problem of Bollobás and Scott

2009
*
Combinatorica
*

The

doi:10.1007/s00493-009-2381-x
fatcat:icalh57sdbdubcrxx2xtildkgi
*bipartite*density of*a**graph*G is max{|E(H)|/|E(G)| : H is*a**bipartite*subgraph of G}. It is NP-hard to determine the*bipartite*density of any*triangle*-*free*cubic*graph*. ...*A*biased maximum*bipartite*subgraph of*a**graph*G is*a**bipartite*subgraph of G with the maximum number of edges such that*one*of its partite sets is independent in G. ... Let G be*a*subcubic*graph*such that each of its blocks is*a**triangle*, or*a*K + 3 , or*a**triangle*-*free**graph*.*Note*that*a*block of G that is*a*K + 3 must be an endblock of G. ...##
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Some spectral inequalities for triangle-free regular graphs

2013
*
Filomat
*

We give three general bounds

doi:10.2298/fil1308561k
fatcat:eif2gkpc4jel5ccgi4nz5dqnfu
*on*the diameter, degree*and*order of*triangle*-*free*regular*graphs*with bounded second largest eigenvalue. ... Next, we consider*bipartite*regular*graphs**and*present another four inequalities that bound the order of such*graphs*in terms of their degree*and*their second largest eigenvalue. ... In both cases second largest eigenvalue is used to check whether*a*given*triangle*-*free*or*triangle*-*free**and*pentagon-*free*regular*graph*is*bipartite*. Corollary 2.1. ...##
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Bipartite density of triangle-free subcubic graphs

2009
*
Discrete Applied Mathematics
*

It was conjectured by Bondy

doi:10.1016/j.dam.2008.07.007
fatcat:d43pemkbf5b23crqewmkq7t2c4
*and*Locke that if G is*a**triangle*-*free*subcubic*graph*, then b(G) ≥ 4 5*and*equality holds only if G is in*a*list of seven small*graphs*. ...*A**graph*is subcubic if its maximum degree is at most 3. The*bipartite*density of*a**graph*G is defined as b(G) = max{|E(B)|/|E(G)| : B is*a**bipartite*subgraph of G}. ... There are*triangle*-*free*subcubic*graphs*with minimum degree 2 that have*bipartite*density 4 Fig. 2 are*triangle*-*free*, subcubic,*and*have*bipartite*density 4 5 . ...
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