{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:45:09Z","timestamp":1787323509030,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"publisher","award":["822\/18"],"award-info":[{"award-number":["822\/18"]}],"id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Given an $n$-vertex graph $H$ with minimum degree at least $d n$ for some fixed $d &gt; 0$, the distribution $H \\cup \\mathbb{G}(n,p)$ over the supergraphs of $H$ is referred to as a (random) perturbation of $H$. We consider the distribution of edge-colored graphs arising from assigning each edge of the random perturbation $H \\cup \\mathbb{G}(n,p)$ a color, chosen independently and uniformly at random from a set of colors of size $r := r(n)$. We prove that edge-colored graphs which are generated in this manner asymptotically almost surely admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies $p := p(n) \\geq C\/n$ for some fixed $C &gt; 0$ and $r = (1 + o(1))n$. The number of colors used is clearly asymptotically best possible. In particular, this improves on a recent result of Anastos and Frieze [ J. Graph Theory, 92 (2019), pp. 405--414] in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-colored sparse pseudorandom graphs asymptotically almost surely admit an almost spanning rainbow path.<\/jats:p>","DOI":"10.1137\/20m1332992","type":"journal-article","created":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T09:30:47Z","timestamp":1626082247000},"page":"1569-1577","source":"Crossref","is-referenced-by-count":9,"title":["Rainbow Hamilton Cycles in Randomly Colored Randomly Perturbed Dense Graphs"],"prefix":"10.1137","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9207-0596","authenticated-orcid":true,"given":"Elad","family":"Aigner-Horev","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8923-3879","authenticated-orcid":true,"given":"Dan","family":"Hefetz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,7,12]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579172"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22461"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20594"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548318000366"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20885"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10112"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10070"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20850"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1112\/mtk.12005"},{"key":"atypb10","first-page":"335","volume":"14","author":"W.","year":"1979","journal-title":"Studia Sci. Math. Hungar."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-2.1.69"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20870"},{"key":"atypb13","first-page":"P1","volume":"21","author":"Ferber A.","year":"2015","journal-title":"Electron. J. Combin."},{"key":"atypb14","first-page":"167","author":"Ferber A.","year":"2016","journal-title":"Berlin"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20475"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"M. Hahn-Klimroth, G. S. Maesaka, Y. Mogge, S. Mohr, and O. Parczyk,\n                      Random Perturbation of Sparse Graphs,\n                      preprint, arXiv:2004.04672, 2020.","DOI":"10.37236\/9510"},{"key":"atypb17","unstructured":"J. Han and Y. Zhao,\n                      Embedding Hamilton $\\ell$-Cycles in Randomly Perturbed Hypergraphs\n                      , preprint, arXiv:1802.04586, 2018."},{"key":"atypb18","first-page":"4","author":"Krivelevich M.","year":"2016","journal-title":"Cambridge"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000079"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/15M1032910"},{"key":"atypb21","first-page":"199","author":"Krivelevich M.","year":"2006","journal-title":"Berlin"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.37236\/7671"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1332992","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:06:28Z","timestamp":1787321188000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1332992"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":22,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1332992"],"URL":"https:\/\/doi.org\/10.1137\/20m1332992","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}