{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:15Z","timestamp":1753893855597,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In 1995, Erd\u00f6s and Gy\u00e1rf\u00e1s proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number $pc_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by $pc_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours?For fixed integers $r&gt;s$ and as $n\\to\\infty$, we prove that $pc_{r,s}(K_n) = \\Theta(n^{1\/\\chi})$, where $\\chi=\\max{\\{1,2+2s-r\\}}$ is the chromatic number of the Kneser graph $KG(r,r-s)$. More generally, if one replaces $K_n$ by an arbitrary $n$-vertex graph with fixed independence number $\\alpha$, then we have $pc_{r,s}(G) = O(n^{1\/\\chi})$, where this time around $\\chi$ is the chromatic number of the Kneser hypergraph $KG^{(\\alpha+1)}(r,r-s)$. This result is tight in the sense that there exist graphs with independence number $\\alpha$ for which $pc_{r,s}(G) = \\Omega(n^{1\/\\chi})$. This is in sharp contrast to the case $r=s$, where it follows from a result of S\u00e1rk\u00f6zy (2012) that $pc_{r,r}(G)$ depends only on $r$ and $\\alpha$, but not on the number of vertices.We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic $d$-regular graphs.<\/jats:p>","DOI":"10.37236\/7469","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:10:43Z","timestamp":1578669043000},"source":"Crossref","is-referenced-by-count":0,"title":["Vertex Covering with Monochromatic Pieces of few Colours"],"prefix":"10.37236","volume":"25","author":[{"given":"Marlo","family":"Eugster","sequence":"first","affiliation":[]},{"given":"Frank","family":"Mousset","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,8,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i3p33\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i3p33\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:26:48Z","timestamp":1579235208000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i3p33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,24]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,7,12]]}},"URL":"https:\/\/doi.org\/10.37236\/7469","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,8,24]]},"article-number":"P3.33"}}