Colourings of star systems

Darijani, Iren; Pike, David A.

doi:10.1002/jcd.21712

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…Originally, the problems in [4] and this paper were to be tools to work toward coloring star systems. For the most part, Darijani and Pike [1] have answered this question. We now hope these results can be used for further research.…”

Section: Discussionmentioning

confidence: 99%

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Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs

Newman

Voloshin

2021

Mathematics

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In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph H that can be placed on a circle and colored according to some rules that are related to colorings of circular mixed hypergraphs. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph of this cycle. We propose an algorithm to color the (r,r)-uniform, complete, circular, mixed hypergraphs for all feasible values with no gaps. In doing so, we show χ(H)=2 and χ¯(H)=n−s or n−s−1 where s is the sieve number.

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Section: Discussionmentioning

confidence: 99%

“…Initially, this paper was inspired by coloring star systems investigated in [1]. Darijani and Pike colored e-stars systems of the complete graph.…”

Section: Introductionmentioning

confidence: 99%

Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs

Newman

Voloshin

2021

Mathematics

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“…More recently, Darijani and Pike [3] showed that for any integer k 2, there exists a k-chromatic 3-star system of order n for all sufficiently large admissible n. They also generalized this result for e-star systems for any e 3. They showed that for all k 2 and e 3, there exists a kchromatic e-star system of order n for all sufficiently large n such that n ≡ 0, 1 (mod 2e).…”

Section: Introductionmentioning

confidence: 87%

“…3 , w 1 , w 3 }, and Y = {v 2 , v 4 , w 2 }. Then B is a decomposition of K 4,3 into P 4 paths for which R and Y constitute an equitable 2-colouring.…”

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confidence: 99%

Colourings of path systems

Darijani¹,

Pike²

2022

Preprint

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A Pm path in a graph is a path on m vertices. A Pm system of order n > 1 is a partition of the edges of the complete graph Kn into Pm paths. A Pm system is said to be k-colourable if the vertex set of Kn 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 Pm 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 observe that there exists a k-chromatic Pm system for any k 2 and m 4 where m is even. Next, we prove that there exists an equitably 2-chromatic P4 system of order n for each admissible order n. We then show that for all k 3, there exists a k-chromatic P4 system of order n for all sufficiently large admissible n. Finally, we show that there exists a uniquely 2-chromatic P4 system of order n for each admissible n 109.

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Colourings of Path Systems

Darijani,

Pike

2023

New Advances in Designs, Codes and Cryptography

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Colourings of star systems

Cited by 4 publications

References 6 publications

Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs

Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs

Colourings of path systems

Colourings of Path Systems

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