Max Cuts in Triangle-Free Graphs

Balogh, József; Clemen, Felix Christian; Lidický, Bernard

doi:10.1007/978-3-030-83823-2_82

Cited by 6 publications

(5 citation statements)

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“…This idea comes from Naves [19] who proposed it at the Graduate Research Workshop in Combinatorics. It was also used by Hu, Lidický, Martins, Norin, and Volec [13] and by the authors in [2].…”

Section: Given a Vertex-subset A We Denote By E(g[a]mentioning

confidence: 99%

“…Another related conjecture of Erdős [6] on triangle-free graphs states that every triangle-free graph on n vertices can be made bipartite by removing at most n 2 /25 edges. There also has been work on this conjecture [2,8,10], with the most recent progress by the authors of this paper who proved that every triangle-free graph on n vertices can be made bipartite by removing at most n 2 /23.5 edges.…”

Section: Introductionmentioning

confidence: 99%

“…r−1 2r (2α − 1 − (1 − α)2 ). Together with (35), this gives((r − 1)k r (1 − α) − s) 2 2r(r − 1) ≤ r − 1 r (1 − α) 2 implying |(r − 1)k r (1 − α) − s| ≤ √ 2(r − 1)(1 − α).…”

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10 Problems for Partitions of Triangle-free Graphs

Balogh¹,

Clemen²,

Lidický³

2022

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We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erdős' Sparse Half Conjecture.Among others we prove the following variant of it: For every sufficiently large even integer n the following holds. Every triangle-free graph on n vertices has a partition16. This result is sharp since the complete bipartite graph with class sizes 3n/4 and n/4 achieves equality, when n is a multiple of 4.Additionally, we discuss similar problems for K 4 -free graphs.

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Section: Given a Vertex-subset A We Denote By E(g[a]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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10 Problems for Partitions of Triangle-free Graphs

Balogh¹,

Clemen²,

Lidický³

2022

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“…There also has been work on this conjecture [1,3,10,12,17], most recently, Balogh, Clemen and Lidický [3] proved D 2 (G) ≤ n 2 /23.5.…”

Section: Introductionmentioning

confidence: 98%

The Spectrum of Triangle-free Graphs

Balogh¹,

Clemen²,

Lidický³

et al. 2022

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Denote by q n (G) the smallest eigenvalue of the signless Laplacian matrix of an n-vertex graph G. Brandt conjectured in 1997 that for regular triangle-free graphs q n (G) ≤ 4n 25 . We prove a stronger result: If G is a triangle-free graph then q n (G) ≤ 15n 94 < 4n 25 . Brandt's conjecture is a subproblem of two famous conjectures of Erdős:(1) Sparse-Half-Conjecture: Every n-vertex triangle-free graph has a subset of vertices of size n 2 spanning at most n 2 /50 edges.(2) Every n-vertex triangle-free graph can be made bipartite by removing at most n 2 /25 edges.In our proof we use linear algebraic methods to upper bound q n (G) by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.

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“…Another question asks what can be the maximum possible ℓ 1 -distance (which in this case is simply the number of edges deleted) from a triangle-free graph to the class of bipartite graphs. It was studied in [EFPS88,EGS92,BCL21].…”

Section: Introductionmentioning

confidence: 99%