Colourings of path systems

TL;DR

This paper explores the colorability properties of path systems in complete graphs, establishing existence results for various chromatic and unique colorability configurations, especially focusing on $P_4$ systems.

Contribution

It proves the existence of $k$-chromatic and uniquely 2-chromatic $P_4$ systems for various orders, advancing understanding of their colorability structures.

Findings

Existence of $k$-chromatic $P_m$ systems for any $k extgreater{}=2$ and even $m extgreater{}=4$.

Existence of equitably 2-chromatic $P_4$ systems for all admissible orders.

Existence of $k$-chromatic $P_4$ systems for large admissible orders and all $k extgreater{}=3$.

Abstract

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 $n \geq 109$.