论文标题
路径系统的着色
Colourings of path systems
论文作者
论文摘要
图表中的$ p_m $路径是$ m $顶点的路径。 $ p_m $ $ n> $ n> 1 $的系统是完整图的边缘$ k_n $的边缘$ p_m $ paths。如果可以将$ k_n $的顶点组分配到$ k $ sets，则可以将$ k $ $ p_m $系统的系统颜色为$ k $ - 颜色，以至于系统中没有单色路径是单色的。如果是$ k $ - 颜色，则该系统为$ k $ chronic，但不是$（k-1）$ - 可颜色。如果每种$ k $ - 颜色$ p_m $ $ $ $ $ $ $ $ $ $ $ ϕ $都可以通过颜色的排列来获得，我们说该系统是唯一的$ k $ - 颜色。在本文中，我们首先观察到，对于任何$ k \ geq 2 $和$ m \ geq 4 $，$ k $ p_m $ $ $ $ $ $ p_m $ $。接下来，我们证明存在一个公平的2-铬$ P_4 $ $ n $的系统$ n $。然后，我们证明，对于所有$ k \ geq 3 $，都存在$ k $ -Chronic $ p_4 $的订单$ n $系统，用于所有足够大的可允许的$ n $。最后，我们证明存在一个独特的2个chrotic $ p_4 $ $ n $的系统，每$ n $ $ n $ n \ geq 109 $。
A $P_m$ path in a graph is a path on $m$ vertices. A $P_m$ system of order $n>1$ is a partition of the edges of the complete graph $K_n$ into $P_m$ paths. A $P_m$ system is said to be $k$-colourable if the vertex set of $K_n$ can be partitioned into $k$ sets called colour classes such that no path in the system is monochromatic. The system is $k$-chromatic if it is $k$-colourable but is not $(k-1)$-colourable. If every $k$-colouring of a $P_m$ 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 observe that there exists a $k$-chromatic $P_m$ system for any $k\geq 2$ and $m\geq 4$ where $m$ is even. Next, we prove that there exists an equitably 2-chromatic $P_4$ system of order $n$ for each admissible order $n$. We then show that for all $k\geq 3$, there exists a $k$-chromatic $P_4$ system of order $n$ for all sufficiently large admissible $n$. Finally, we show that there exists a uniquely 2-chromatic $P_4$ system of order $n$ for each admissible $n \geq 109$.
