I suppose German Zigeuner and Romanian ţigan (gypsy) share a common etymology from Greek τσιγγάνος. I wonder if Swedish tigga, which means to beg and looks/sounds quite similar to ţigan, has an etymological link. Maybe people historically associated begging with gypsies, so that the Swedish word for begging originated from the same source. Does it?
The answer is no.
At first, not everyone agrees on the etymology of this ethnonym, or, rather, an exonym. Both Duden-7 and Kluge say that the origin of German Zigeuner is "ungewiss" (uncertain) or "unklar" (not clear). Duden online says "Herkunft ungeklärt". The authors of Etymologisch Woordenboek van het Nederlands are of the same opinion re: Dutch "zigeuner" - "waarvan de herkomst echter onduidelijk is". Anyhow, this word has been in use in German (and in most other languages spoken in Europe) since the 15th century. Also note that you might not want to use it in most situations, cf. Duden Online's remark that "Die Bezeichnung Zigeuner, Zigeunerin wird vom Zentralrat Deutscher Sinti und Roma als diskriminierend abgelehnt."
Vasmer does indeed argue that it comes from Medieval (Byzantine) Greek, citing Theophanes' Chronographia, who described them as "haeretici in Phrygia et Lycaonia praecipue degentes". Some argue it is related to Ancient Greek θιγγάνω "touch" and translate the ethnonym as "untouched". My knowledge of Ancient Greek morphology is not good enough to critically evaluate this claim.
Now to a possible link between Swedish "tigga" and this ethnonym. The word "tigga" is a rather old word. It is related to OE þicg(e)an "to take, receive, accept; esp. to take (food), to consume by eating or drinking"; see other comparanda; also note that the authors of Svenska Akademiens ordbok argue that the origin of Swedish tigga is "av ovisst ursprung" - not sure why. Anyhow, it definitely predates Swedish "zigenare" (note the following from an article in Swedish wikipedia "Både termen gypsy och termen zigenare anses av en del romer vara oacceptabla benämningar på dem.")
The words for Gypsy are as follows, zingari/zinger/tinker, cigany,atzigano/gitano. Take a look at a book called The Secrets of Society (by Dai'Quiriya Martinez- available at Amazon.com). Many of these are terms for Gypsies that originate in the Middle Ages in Europe and some of these words actually refer to the dark skin color of the Roma (Gypsy) ethnic group. The root for the words zingari/ zinger is zinj. The word zinj has its origins in the word in the name Zanj/Zinj, is from Arabic origins and refers to East Africa.