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- Title
Colourings of star systems.
- Authors
Darijani, Iren; Pike, David A.
- Abstract
An e‐star is a complete bipartite graph K1,e. An e‐star system of order n>1,Se(n), is a partition of the edges of the complete graph Kn into e‐stars. An e‐star system is said to be k‐colourable if its vertex set can be partitioned into k sets (called colour classes) such that no e‐star is monochromatic. The system Se(n) is k‐chromatic if Se(n) is k‐colourable but is not (k−1)‐colourable. If every k‐colouring of an e‐star 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 show that for any integer k⩾2, there exists a k‐chromatic 3‐star system of order n for all sufficiently large admissible n. Next, we generalize this result for e‐star systems for any e⩾3. We show that for all k⩾2 and e⩾3, there exists a k‐chromatic e‐star system of order n for all sufficiently large n such that n≡0,1 (mod 2e). Finally, we prove that for all k⩾2 and e⩾3, there exists a uniquely k‐chromatic e‐star system of order n for all sufficiently large n such that n≡0,1 (mod 2e).
- Subjects
BIPARTITE graphs; COMPLETE graphs; COLOR; PERMUTATIONS; RAMSEY theory
- Publication
Journal of Combinatorial Designs, 2020, Vol 28, Issue 7, p525
- ISSN
1063-8539
- Publication type
Article
- DOI
10.1002/jcd.21712