Large Monochromatic Triple Stars in Edge Colourings

Shoham Letzter

doi:10.1002/jgt.21854

Minimum Number of Monotone Subsequences of Length 4 in Permutations

Combinatorics Probability Computing ◽

10.1017/s0963548314000820 ◽

2014 ◽

Vol 24 (4) ◽

pp. 658-679 ◽

Cited By ~ 11

Author(s):

JÓZSEF BALOGH ◽

PING HU ◽

BERNARD LIDICKÝ ◽

OLEG PIKHURKO ◽

BALÁZS UDVARI ◽

...

Keyword(s):

Lower Bound ◽

Complete Graphs ◽

Formula Group ◽

Minimum Number ◽

Additional Stability ◽

Edge Colourings ◽

Image Position

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

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A Canonical Ramsey Theorem for Exactly m-Coloured Complete Subgraphs

Combinatorics Probability Computing ◽

10.1017/s0963548313000503 ◽

2013 ◽

Vol 23 (1) ◽

pp. 102-115 ◽

Cited By ~ 3

Author(s):

TEERADEJ KITTIPASSORN ◽

BHARGAV P. NARAYANAN

Keyword(s):

Natural Number ◽

Complete Graph ◽

Infinite Subset ◽

Complete Subgraph ◽

Ramsey Theorem ◽

Distinguished Vertex ◽

Edge Colourings ◽

Distinct Colour

Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ $\mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex v in X such that X\{v} is 1-coloured and each edge between v and X\{v} has a distinct colour (all different to the colour used on X\{v}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

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Canonical edge-colourings of locally finite graphs

COMBINATORICA ◽

10.1007/bf02579280 ◽

1982 ◽

Vol 2 (1) ◽

pp. 37-51 ◽

Cited By ~ 9

Author(s):

A. J. W. Hilton

Keyword(s):

Locally Finite ◽

Finite Graphs ◽

Locally Finite Graphs ◽

Edge Colourings

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Characterisation Results for Steiner Triple Systems and Their Application to Edge-Colourings of Cubic Graphs

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-021-9 ◽

2010 ◽

Vol 62 (2) ◽

pp. 355-381 ◽

Cited By ~ 5

Author(s):

Daniel Král’ ◽

Edita Máčajov´ ◽

Attila Pór ◽

Jean-Sébastien Sereni

Keyword(s):

Triple System ◽

Steiner Triple System ◽

Cubic Graphs ◽

Steiner Triple Systems ◽

Cubic Graph ◽

Triple Systems ◽

Forbidden Configurations ◽

Edge Colourings

AbstractIt is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration C14. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations.Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph G is S-edge-colourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are S-edge-colourable for every non-projective nonaffine point-transitive Steiner triple system S.

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On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

The Electronic Journal of Combinatorics ◽

10.37236/5173 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 8

Author(s):

Jakub Przybyło

Keyword(s):

Connected Graph ◽

Special Class ◽

Maximum Degree ◽

Minimum Degree ◽

Probabilistic Approach ◽

The Family ◽

Edge Colourings ◽

Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

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