Colourings of star systems

Abstract

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 $ϕ$ 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 $2e$).