论文标题
恒星系统的块
Block colourings of star systems
论文作者
论文摘要
$ e $ -Star的订单$ n $的系统是完整的图形$ k_n $的分解，以完成完整的两部分图$ k_ {1，e} $（或$ e $ -star）。众所周知，仅当$ n \ geq 2e $和$ e $ divides $ \ binom {n} {2} $时，才知道此类系统存在。我们考虑了此类系统的块着色，其中每个$ e $ - 标准都被分配了颜色，而共享顶点的两个$ e $标准会获得不同的颜色。我们介绍了针对小$ 3 $标准系统的块颜色的计算机分析。此外，我们证明：（i）对于$ n \ equiv 0,1 $ mod $ 2e $，存在$ n $或$（n-1）$ - 块可着色$ e $ e $ e $ -Star System of Order $ n $; （ii）当$ e = 3 $时，剩余的一致性类别mod $ 6 $都会产生相同的结果。
An $e$-star system of order $n$ is a decomposition of the complete graph $K_n$ into copies of the complete bipartite graph $K_{1,e}$ (or $e$-star). Such systems are known to exist if and only if $n\geq 2e$ and $e$ divides $\binom{n}{2}$. We consider block colourings of such systems, where each $e$-star is assigned a colour, and two $e$-stars which share a vertex receive different colours. We present a computer analysis of block colourings of small $3$-star systems. Furthermore, we prove that: (i) for $n\equiv 0,1$ mod $2e$ there exists either an $n$ or $(n-1)$-block colourable $e$-star system of order $n$; and (ii) when $e=3$, the same result holds in the remaining congruence classes mod $6$.
