{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:14:49Z","timestamp":1758824089885,"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>It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\\delta(G)\\ge{3n\\over 4}$ ensures a monochromatic connected component with at least $\\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is\u00a0 that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\\over k-1}$, in fact $$\\delta(G)\\ge \\left(1 - \\frac{1}{1000(k-1)^9}\\right)n$$ suffices.Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with\u00a0 $\\delta(G)\\geq {9n\\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\\over 2}$. We conjecture that this can be improved to ${7n\\over 9}$ and for general $k$ we conjecture the following: if $k\\geq 3$ and\u00a0 $G$ is a graph of order $n$ such that $\\delta(G)\\geq \\left( 1 - \\frac{k-1}{k^2}\\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\\over k-1}$.<\/jats:p>","DOI":"10.37236\/7049","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:45:56Z","timestamp":1578671156000},"source":"Crossref","is-referenced-by-count":5,"title":["Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition"],"prefix":"10.37236","volume":"24","author":[{"given":"Andr\u00e1s","family":"Gy\u00e1rf\u00e1s","sequence":"first","affiliation":[]},{"given":"G\u00e1bor","family":"S\u00e1rk\u00f6zy","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,9,8]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p54\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p54\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:46:08Z","timestamp":1579236368000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,8]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/7049","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,9,8]]},"article-number":"P3.54"}}