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Colourings of star systems
Journal of Combinatorial Designs ( IF 0.5 ) Pub Date : 2020-03-28 , DOI: 10.1002/jcd.21712
Iren Darijani 1 , David A. Pike 1
Affiliation  

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 $2e$).

中文翻译:

恒星系统的着色

$e$-star 是一个完整的二部图 $K_{1,e}$。一个$n>1$ 阶$e$-star 系统,$S_e(n)$,是将完整图$K_n$ 的边划分为$e$-stars。如果 $e$-star 系统的顶点集可以划分为 $k$ 集(称为颜色类),使得没有 $e$-star 是单色的,则称 $e$-star 系统是 $k$-可着色的。如果 $S_e(n)$ 是 $k$-colourable 但不是 $(k-1)$-colourable,则系统 $S_e(n)$ 是 $k$-colorable。如果 $e$-star 系统的每个 $k$-colouring 都可以通过颜色的排列从一些 $k$-colouring $\phi$ 中获得,我们就说该系统是唯一的 $k$-colourable。在本文中,我们首先证明对于任何整数 $k\geq 2$,对于所有足够大的可容许 $n$,都存在一个 $k$-色阶 $n$ 的三星系统。接下来,我们将这个结果推广到任何 $e\geq 3$ 的 $e$-star 系统。我们证明,对于所有 $k\geq 2$ 和 $e\geq 3$,对于所有足够大的 $n$ 都存在一个 $k$-色的 $e$-star 系统,其阶 $n$ 使得 $n\相当于 0,1$(mod $2e$)。最后,我们证明对于所有的 $k\geq 2$ 和 $e\geq 3$,对于所有足够大的 $n$ 都存在一个唯一的 $k$-色度 $e$-star 系统,其阶 $n$ 使得$n\equiv 0,1$ (mod $2e$)。
更新日期:2020-03-28
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