Colouring lines in projective space

Chowdhury, Ameera; Godsil, Chris; Royle, Gordon F.

doi:10.1016/j.jcta.2005.01.010

Cited by 16 publications

(14 citation statements)

References 7 publications

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“…Two vertices of qK n:k are adjacent if and only if the corresponding k-subspaces are disjoint. In [3], the chromatic number of the q-Kneser graph qK n:2 is determined, and the minimum colorings are characterized. In [18], the chromatic number of the q-Kneser graph is determined in general for q > q k .…”

Section: The Electronic Journal Of Combinatorics 17 (2010) #R71mentioning

confidence: 99%

See 1 more Smart Citation

A Hilton-Milner Theorem for Vector Spaces

Blokhuis¹,

Brouwer²,

Chowdhury³

et al. 2010

Electron. J. Combin.

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We show for $k \geq 2$ that if $q\geq 3$ and $n \geq 2k+1$, or $q=2$ and $n \geq 2k+2$, then any intersecting family ${\cal F}$ of $k$-subspaces of an $n$-dimensional vector space over $GF(q)$ with $\bigcap_{F \in {\cal F}} F=0$ has size at most $\left[{n-1\atop k-1}\right]-q^{k(k-1)}\left[{n-k-1\atop k-1}\right]+q^k$. This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chromatic number of the corresponding $q$-Kneser graphs.

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Section: The Electronic Journal Of Combinatorics 17 (2010) #R71mentioning

confidence: 99%

“…It remains to settle the case n = 7, k = l = 3, and q 3. By Lemma 2.4, we can choose a 1-space E such that |F E | |F |/ 3 1 and a 2-space S on E such that |F…”

Section: The Case L = Kmentioning

confidence: 99%

A Hilton-Milner Theorem for Vector Spaces

Blokhuis¹,

Brouwer²,

Chowdhury³

et al. 2010

Electron. J. Combin.

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“…For k = 2, the chromatic number was determined earlier by Chowdhury et al [14]. For that case they proved χ(qK n:k ) = n−1 1…”

Section: Moreover Equality Holds If and Only If F = {F ∈ V Kmentioning

confidence: 82%

On q-analogues and stability theorems

et al. 2011

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Abstract. In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.Mathematics Subject Classification (2010). 05D05, 05B25.

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“…If L(x i ) is the set of lines incident with x i , it follows that the sets L(x i ) provide a covering of the Kneser graph of lines by independent sets. These Kneser graphs have been studied in [5] where it was shown that also they have chromatic number q 2 + q. Clearly every coloring of the Kneser graph of lines of PG(3, q) provides a coloring of the graph of chambers studied in the present paper, by replacing each color class C by the set of all chambers whose line lies in C. Therefore the optimal colorings in both graphs correspond to one another and therefore we refer to [5] for further information.…”

Section: A Stability Resultsmentioning

confidence: 99%

The Chromatic Number of Two Families of Generalized Kneser Graphs Related to Finite Generalized Quadrangles and Finite Projective 3-Spaces

Metsch¹

2021

Electron. J. Combin.

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Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective space $\mathrm{PG}(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\geqslant 43$ we determine the largest independent set of $\Gamma$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\geqslant 47$, this information is then used to show that $\Gamma$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove similar results for the graph that is built in the same way on the chambers of the generalized quadrangle.

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Colouring lines in projective space

Cited by 16 publications

References 7 publications

A Hilton-Milner Theorem for Vector Spaces

A Hilton-Milner Theorem for Vector Spaces

On q-analogues and stability theorems

The Chromatic Number of Two Families of Generalized Kneser Graphs Related to Finite Generalized Quadrangles and Finite Projective 3-Spaces

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