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• W2145527772 abstract "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 $Bin{cal B}$, let $phi(B)={phi(x) | xin B}$. 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 $c$ (${times}{times}{square}{square}$, 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 $m=2$." @default.
• W2145527772 created "2016-06-24" @default.
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• W2145527772 date "2001-06-05" @default.
• W2145527772 modified "2023-10-14" @default.
• W2145527772 title "Colouring $4$-cycle Systems with Specified Block Colour Patterns: the Case of Embedding $P_3$-designs" @default.
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• W2145527772 doi "https://doi.org/10.37236/1568" @default.
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