K_4-free graphs have sparse halves

Reiher, Christian

doi:10.48550/arxiv.2108.07297

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K 4 -Free Graphs2

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2022

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Cited by 1 publication

(2 citation statements)

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“…The following conjecture, if true, were to generalize both Sudakov's and Reiher's result. Note that Reiher's result [23] implies Conjecture 2.4 for regular graphs. Further, it is also true for 3-partite graphs.…”

Section: K 4 -Free Graphsmentioning

confidence: 95%

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10 Problems for Partitions of Triangle-free Graphs

Balogh¹,

Clemen²,

Lidický³

2022

Preprint

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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: K 4 -Free Graphsmentioning

confidence: 95%

“…Recently Reiher [23], building up on work of Liu and Ma [18], proved the corresponding sparsehalf-version of this theorem: Every K 4 -free graphs contains a set of size n/2 spanning at most n 2 /18 edges. The following conjecture, if true, were to generalize both Sudakov's and Reiher's result.…”

Section: K 4 -Free Graphsmentioning

confidence: 99%