Classification of Edge-Transitive Rose Window Graphs Istv ´ an Kov ´ acs, 1 Klavdija Kutnar, 1 and Dragan Maru ˇ si ˇ c 1,2 1 UNIVERSITY OF PRIMORSKA, FAMNIT GLAGOLJA ˇ SKA 8, 6000 KOPER, SLOVENIA E-mail: istvan.kovacs@upr.si; klavdija.kutnar@upr.si; dragan.marusic@upr.si 2 UNIVERSITY OF LJUBLJANA, IMFM JADRANSKA 19, 1000 LJUBLJANA, SLOVENIA Received 12 September 2008; Revised 8 October 2009 Published online 22 January 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jgt.20475 Abstract: Given natural numbers n ≥ 3 and 1 ≤ a, r ≤ n - 1, the rose window graph R n (a, r ) is a quartic graph with vertex set {x i |i ∈ Z n }∪{y i |i ∈ Z n } and edge set {{x i , x i +1 }| i ∈ Z n }∪{{y i , y i +r }| i ∈ Z n }∪{{x i , y i }| i ∈ Z n }∪ {{x i +a , y i }| i ∈ Z n }. In this article a complete classification of edge-transitive rose window graphs is given, thus solving one of the three open problems about these graphs posed by Steve Wilson in 2001.  2010 Wiley Periodicals, Inc. J Graph Theory 65: 216–231, 2010 Keywords: group; graph; rose window; vertex-transitive; edge-transitive; arc-transitive Contract/grant sponsor: Agencija za raziskovalno dejavnost Republike Slovenije; Contract/grant number: P1-0285. Journal of Graph Theory  2010 Wiley Periodicals, Inc. 216