{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:02Z","timestamp":1753893782842,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $c$ (${\\times}{\\times}{\\square}{\\square}$, 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 $m=2$.<\/jats:p>","DOI":"10.37236\/1568","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:07:27Z","timestamp":1578708447000},"source":"Crossref","is-referenced-by-count":6,"title":["Colouring $4$-cycle Systems with Specified Block Colour Patterns: the Case of Embedding $P_3$-designs"],"prefix":"10.37236","volume":"8","author":[{"given":"Gaetano","family":"Quattrocchi","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2001,6,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r24\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v8i1r24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:17:54Z","timestamp":1579324674000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v8i1r24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,6,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2001,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1568","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2001,6,5]]},"article-number":"R24"}}