Block colourings of star systems

TL;DR

This paper investigates the coloring properties of star system decompositions of complete graphs, providing existence results for certain colorings based on modular conditions and analyzing small cases computationally.

Contribution

It introduces new existence results for block colorings of $e$-star systems under specific modular conditions, including computational analysis for small 3-star systems.

Findings

Existence of $n$ or $(n-1)$-block colorable $e$-star systems for certain congruences.

Computer analysis of small 3-star systems' colorings.

Coloring existence results depend on modular arithmetic conditions.

Abstract

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 $6$.