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

Gaetano Quattrocchi
2001 Electronic Journal of Combinatorics
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 $c$
more » ... \times}{\times}{\square}{\square}$, two-coloured of pattern$2+2$), type$d$(${\times}{\square}{\times}{\square}$, mixed two-colored), type$e$(${\times}{\times}{\square}{\triangle}$, three-coloured of pattern$2+1+1$), type$f$(${\times}{\square}{\times}{\triangle}$, mixed three-coloured), type$g$(${\times}{\square}{\triangle}{\diamondsuit}$, four-coloured or polychromatic).Let$S$be a subset of$\{a,b,c,d,e,f,g\}$. An$m$-colouring$\phi$of$(V,{\cal B})$is said of type$S$if the type of every$4$-cycle of$\cal B$is in$S$. A type$S$colouring is said to be proper if for every type$\alpha \in S$there is at least one$4$-cycle of$\cal B$having colour type$\alpha$.We say that a$P(v,3,1)$,$(W,{\cal P})$, is embedded in a$4$-cycle system of order$n$,$(V,{\cal B})$, if every path$p=[a_1,a_2,a_3] \in {\cal P}$occurs in a$4$-cycle$(a_1,a_2,a_3,x) \in {\cal B}$such that$x \notin W$.In this paper we consider the following spectrum problem: given an integer$m$and a set$S \subseteq \{b,d,f\}$, determine the set of integers$n$such that there exists a$4$-cycle system of order$n$with a proper$m$-colouring of type$S$(note that each colour class of a such coloration is the point set of a$P_3$-design embedded in the$4$-cycle system).We give a complete answer to the above problem except when$S=\{b\}$. In this case the problem is completely solved only for$m=2\$.