The chromatic thresholds of graphs

Chromatic thresholds in sparse random graphs

Random Structures and Algorithms ◽

10.1002/rsa.20709 ◽

2017 ◽

Vol 51 (2) ◽

pp. 215-236

Author(s):

Peter Allen ◽

Julia Böttcher ◽

Simon Griffiths ◽

Yoshiharu Kohayakawa ◽

Robert Morris

Keyword(s):

Random Graphs ◽

Chromatic Thresholds

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ON THE SATURATIONS AND CHROMATIC THRESHOLDS OF THE SPECTRAL COLOURS1

British Journal of Psychology General Section ◽

10.1111/j.2044-8295.1931.tb00592.x ◽

1931 ◽

Vol 21 (3) ◽

pp. 283-313 ◽

Cited By ~ 2

Author(s):

D. McL. PURDY

Keyword(s):

Chromatic Thresholds

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On the Chromatic Thresholds of Hypergraphs

Combinatorics Probability Computing ◽

10.1017/s0963548315000061 ◽

2015 ◽

Vol 25 (2) ◽

pp. 172-212

Author(s):

JÓZSEF BALOGH ◽

JANE BUTTERFIELD ◽

PING HU ◽

JOHN LENZ ◽

DHRUV MUBAYI

Keyword(s):

Chromatic Number ◽

Minimum Degree ◽

Directed Graphs ◽

Open Problems ◽

Fibre Bundles ◽

Uniform Hypergraphs ◽

The Family ◽

Turán Number ◽

Chromatic Thresholds ◽

Extremal Set

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

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