TY  - JOUR
ID  - Bonacini2017
LB  - Bonacini2017
AU  - Bonacini, Paola
AU  - Gionfriddo, Mario
AU  - Marino, Lucia
TI  - Block colourings of 6-cycle systems
ST  - Block colourings of 6-cycle systems
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2017
VL  - 37
IS  - 5
SP  - 647
EP  - 664
DO  - http://dx.doi.org/10.7494/OpMath.2017.37.5.647
UR  - http://dx.doi.org/10.7494/OpMath.2017.37.5.647
AB  - 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 \(c\).
KW  - 6-cycles
KW  - block-colourings
KW  - G-decompositions
SN  - 1232-9274
ER  -