| Autor: |
Darijani, Iren, Pike, David A. |
| Rok vydání: |
2022 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| DOI: |
10.1007/978-3-031-48679-1_6 |
| Popis: |
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 $\phi$ 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 $n \geq 109$. |
| Databáze: |
arXiv |
| Externí odkaz: |
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