Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Colourings of star systems   An $e$-star is a complete bipartite graph $K_{1,e}$. An $e$-star system of
order $n>1$, $S_e(n)$, is a partition of the edges of the complete graph $K_n$
into $e$-stars. An $e$-star system is said to be $k$-colourable if its vertex
set can be partitioned into $k$ sets (called colour classes) such that no
$e$-star is monochromatic. The system $S_e(n)$ is $k$-chromatic if $S_e(n)$ is
$k$-colourable but is not $(k-1)$-colourable. If every $k$-colouring of an
$e$-star system can be obtained from some $k$-colouring $\phi$ by a permutation
of the colours, we say that the system is uniquely $k$-colourable. In this
paper, we first show that for any integer $k\geq 2$, there exists a
$k$-chromatic 3-star system of order $n$ for all sufficiently large admissible
$n$. Next, we generalize this result for $e$-star systems for any $e\geq 3$. We
show that for all $k\geq 2$ and $e\geq 3$, there exists a $k$-chromatic
$e$-star system of order $n$ for all sufficiently large $n$ such that $n\equiv
0,1$ (mod $2e$). Finally, we prove that for all $k\geq 2$ and $e\geq 3$, there
exists a uniquely $k$-chromatic $e$-star system of order $n$ for all
sufficiently large $n$ such that $n\equiv 0,1$ (mod $2e$).

1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: stat