
TL;DR
This paper proves the existence of k-chromatic and uniquely k-chromatic e-star systems of large order n, for all sufficiently large n satisfying certain modular conditions, generalizing previous results for 3-star systems.
Contribution
It establishes the existence of k-chromatic and uniquely k-chromatic e-star systems for all large admissible n, extending known results from 3-star to e-star systems for e ≥ 3.
Findings
Existence of k-chromatic 3-star systems for large n.
Generalization to e-star systems for e ≥ 3.
Existence of uniquely k-chromatic e-star systems for large n.
Abstract
An -star is a complete bipartite graph . An -star system of order , , is a partition of the edges of the complete graph into -stars. An -star system is said to be -colourable if its vertex set can be partitioned into sets (called colour classes) such that no -star is monochromatic. The system is -chromatic if is -colourable but is not -colourable. If every -colouring of an -star system can be obtained from some -colouring by a permutation of the colours, we say that the system is uniquely -colourable. In this paper, we first show that for any integer , there exists a -chromatic 3-star system of order for all sufficiently large admissible . Next, we generalize this result for -star systems for any . We show that for all and , there exists a…
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