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A note on triangle-free and bipartite graphs

Hans Jürgen Prömel, Thomas Schickinger, Angelika Steger
2002 Discrete Mathematics  
Using 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.  ... 
doi:10.1016/s0012-365x(02)00511-3 fatcat:dwwha7zyd5g27dwbaymgphmv4q

Many disjoint triangles in co-triangle-free graphs [article]

Mykhaylo Tyomkyn
2020 arXiv   pre-print
Our results answer 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.  ... 
arXiv:2001.00763v2 fatcat:5wm6wn7slvhvhegqv36alyonby

A note on triangle-free graphs [article]

Vahan V. Mkrtchyan, Petros A. Petrosyan
2011 arXiv   pre-print
We show that if G is 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.  ... 
arXiv:1101.3188v1 fatcat:5ygextdwhjfblkek4zkmnoe6v4

Triangle-Free Subgraphs of Random Graphs

Peter Allen, Julia Böttcher, Barnaby Roberts, Yoshiharu Kohayakawa
2015 Electronic Notes in Discrete Mathematics  
The Andrásfai-Erdős-Sós Theorem [2] states that all 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.  ... 
doi:10.1016/j.endm.2015.06.055 fatcat:wvqdi4j3knd7bfyrn7tkmgafoq

A note on bipartite subgraphs and triangle-independent sets

Honghai Xu
2017 Discrete Mathematics  
Let α_1 (G) denote the maximum size of an edge set that contains at most 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.  ... 
doi:10.1016/j.disc.2016.07.021 fatcat:ndzg3t36hncdvjhy3ju3uvoi5e

Dense induced bipartite subgraphs in triangle-free graphs [article]

Matthew Kwan, Shoham Letzter, Benny Sudakov, Tuan Tran
2019 arXiv   pre-print
The problem of finding dense induced 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 }.  ... 
arXiv:1810.12144v3 fatcat:rlzpgnnh5nadvhokvowyl5vvbu

Euler Graphs, Triangle-Free Graphs and Bipartite Graphs in Switching Classes [chapter]

Jurriaan Hage, Tero Harju, Emo Welzl
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.  ... 
doi:10.1007/3-540-45832-8_13 fatcat:6oz35fbvfrfn3hftjqj2cgvesy

Bipartite induced density in triangle-free graphs [article]

Wouter Cames van Batenburg, Rémi de Joannis de Verclos, Ross J. Kang, François Pirot
2020 arXiv   pre-print
We prove that any 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.  ... 
arXiv:1808.02512v3 fatcat:lvzw7566tbgjnhmdt26lmubgty

The Graph Tessellation Cover Number: Extremal Bounds, Efficient Algorithms and Hardness [chapter]

Alexandre Abreu, Luís Cunha, Tharso Fernandes, Celina de Figueiredo, Luis Kowada, Franklin Marquezino, Daniel Posner, Renato Portugal
2018 Lecture Notes in Computer Science  
We establish upper bounds 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).  ... 
doi:10.1007/978-3-319-77404-6_1 fatcat:ufrymdqywze2jozmfuh6w33ifi

Bipartite Induced Density in Triangle-Free Graphs

Wouter Cames van Batenburg, Rémi De Joannis de Verclos, Ross J. Kang, François Pirot
2020 Electronic Journal of Combinatorics  
We prove that any 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.  ... 
doi:10.37236/8650 fatcat:gkhy6dapx5c7hh4r7ql5hqekum

Separation Choosability and Dense Bipartite Induced Subgraphs

Louis Esperet, Ross Kang, Stéphan Thomassé
2019 Combinatorics, probability & computing  
For example, does every 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  ... 
doi:10.1017/s0963548319000026 fatcat:yjnunsgmvvcstlci77s3dou4hy

Triangle-free graphs and forbidden subgraphs

Stephan Brandt
2002 Discrete Applied Mathematics  
In particular, we introduce chordal 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 .  ... 
doi:10.1016/s0166-218x(01)00277-3 fatcat:2v7gp75pcvaozdcdmucgaiplpm

On a bipartition problem of Bollobás and Scott

Baogang Xu, Xingxing Yu
2009 Combinatorica  
The 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.  ... 
doi:10.1007/s00493-009-2381-x fatcat:icalh57sdbdubcrxx2xtildkgi

Some spectral inequalities for triangle-free regular graphs

Tamara Koledin, Zoran Stanic
2013 Filomat  
We give three general bounds 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.  ... 
doi:10.2298/fil1308561k fatcat:eif2gkpc4jel5ccgi4nz5dqnfu

Bipartite density of triangle-free subcubic graphs

Xuding Zhu
2009 Discrete Applied Mathematics  
It was conjectured by Bondy 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 .  ... 
doi:10.1016/j.dam.2008.07.007 fatcat:d43pemkbf5b23crqewmkq7t2c4
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