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Colouring $4$-cycle Systems with Specified Block Colour Patterns: the Case of Embedding $P_3$-designs

2001
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Electronic Journal of Combinatorics
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A colouring of a $4$-cycle system $(V,{\cal B})$ is a surjective mapping $\phi : V \rightarrow \Gamma$. The elements of $\Gamma$ are colours. If $|\Gamma|=m$, we have an $m$-colouring of $(V,{\cal B})$. For every $B\in{\cal B}$, let $\phi(B)=\{\phi(x) | x\in B\}$. There are seven distinct colouring patterns in which a $4$-cycle can be coloured: type $a$ (${\times}{\times}{\times}{\times}$, monochromatic), type $b$ (${\times}{\times}{\times}{\square}$, two-coloured of pattern $3+1$), type $c$

doi:10.37236/1568
fatcat:iriobpu4frg5rfvumajcfl5w7a