Opuscula Mathematica
Opuscula Math. 37, no. 5 (2017), 647-664

Block colourings of 6-cycle systems

Paola Bonacini
Mario Gionfriddo
Lucia Marino
Abstract. Let \(\Sigma=(X,\mathcal{B})\) be a \(6\)-cycle system of order \(v\), so \(v\equiv 1,9\mod 12\). A \(c\)-colouring of type \(s\) is a map \(\phi\colon\mathcal {B}\rightarrow \mathcal{C}\), with \(C\) set of colours, such that exactly \(c\) colours are used and for every vertex \(x\) all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(\frac{v-1}{2}=qs+r\), with \(q, r\geq 0\). \(\phi\) is equitable if for every vertex \(x\) the set of the \(\frac{v-1}{2}\) blocks containing \(x\) is partitioned in \(r\) colour classes of cardinality \(q+1\) and \(s-r\) colour classes of cardinality \(q\). In this paper we study bicolourings and tricolourings, for which, respectively, \(s=2\) and \(s=3\), distinguishing the cases \(v=12k+1\) and \(v=12k+9\). In particular, we settle completely the case of \(s=2\), while for \(s=3\) we determine upper and lower bounds for \(c\).
Keywords: 6-cycles, block-colourings, G-decompositions.
Mathematics Subject Classification: 05C15, 05B05.
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About the authors ▼
Paola Bonacini
Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy
Mario Gionfriddo
Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy
Lucia Marino
Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania, Italy
About the article ▼
Communicated by Mariusz Meszka.
Received: 2016-09-19.
Revised: 2016-12-10.
Accepted: 2016-12-21.
Published online: 2017-07-05.
Cite this article as ▼
Paola Bonacini, Mario Gionfriddo, Lucia Marino, Block colourings of 6-cycle systems, Opuscula Math. 37, no. 5 (2017), 647-664, http://dx.doi.org/10.7494/OpMath.2017.37.5.647.
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