The Chromatic Number of the $q$-Kneser Graph for Large $q$

Ihringer, Ferdinand

doi:10.37236/7907

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…Note that [22,Theorem 1.4] requires that k < q log q − q − 1. This condition can be removed, see [18,Theorem 1.8]. Complementary to the two previous results, Theorem 1 also implies that the situation is known for small q.…”

Section: Introductionmentioning

confidence: 52%

Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$

D'haeseleer¹,

Ihringer²,

Mannaert

et al. 2021

Electron. J. Combin.

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We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\mathrm{AG}(n,q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\mathrm{PG}(n,q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\mathrm{AG}(n,q)$ correspond to certain equitable bipartitions of the association scheme of $k$-spaces in $\mathrm{AG}(n,q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\mathrm{AG}(n,q)$. We define Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ and $\mathrm{PG}(n,q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets. This paper focuses on $\mathrm{AG}(n,q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\mathrm{AG}(3,q)$ was already treated separately.

show abstract

Section: Introductionmentioning

confidence: 52%

Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$

D'haeseleer¹,

Ihringer²,

Mannaert

et al. 2021

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Note that [22,Theorem 1.4] requires that k < q log q − q − 1. This condition can be removed, see [18,Theorem 1.8]. Complementary to the two previous results, Theorem 1.1 also implies that the situation is known for small q.…”

Section: Introductionmentioning

confidence: 56%

Cameron-Liebler $k$-sets in $\text{AG}(n,q)$

D'haeseleer,

Ihringer,

Mannaert

et al. 2020

Preprint

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We study Cameron-Liebler k-sets in the affine geometry, so sets of k-spaces in AG (n, q). This generalizes research on Cameron-Liebler k-sets in the projective geometry PG(n, q). Note that in algebraic combinatorics, Cameron-Liebler k-sets of AG(n, q) correspond to certain equitable bipartitions of the association scheme of k-spaces in AG(n, q), while in the analysis of Boolean functions, they correspond to Boolean degree 1 functions of AG(n, q).We define Cameron-Liebler k-sets in AG(n, q) by intersection properties with kspreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler k-sets in AG(n, q) and PG(n, q). As a byproduct, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler k-sets.This paper focuses on AG(n, q) for n > 3, while the case for Cameron-Liebler line classes in AG(3, q) was already treated separately.

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“…Of particular interest in what follows, will be the q-analog of the Kneser graph, denoted qK n:m , whose set of vertices is [ V m ], where V = F n q , and an undirected edge connects two vertices iff the corresponding m-dimensional subspaces have a trivial intersection (e.g., see [6], [7], [15] and references therein).…”

Section: Minimal Network With Two Source Messagesmentioning

confidence: 99%