Origin of fag (meaning a cigarette in British English)

Solution 1:

According to the Online Etymology Dictionary, it comes from "fag", meaning a loose piece of cloth:

fag (n.1) Look up fag at Dictionary.com British slang for "cigarette" (originally, especially, the butt of a smoked cigarette), 1888, probably from fag "loose piece, last remnant of cloth" (late 14c., as in fag-end "extreme end, loose piece," 1610s), which perhaps is related to fag (v.), which could make it a variant of flag (v.).

The OED is a bit more convinced and categorically links it to fag-end:

Etymology: Abbreviation of fag-end. (Cf. fag sb.2 2.)

a. The fag-end of a cigarette. b. A cheap cigarette. c. Any cigarette (the current use). Also attrib., as fag card, a cigarette card; fag hag (see quot. 1945).

• 1888 Sat. Rev. 30 June 786/2 ― They··burn their throats with the abominable ‘fag’, with its acrid paper and vile tobacco.
• 1893 Pick-me-up 14 Oct. 45/2 ― Stimulants he calls ‘booze’ and a cigarette a ‘fag’.
• 1898 Eng. Dial. Dict. s.v., ― Here [sc. at Redruth] we are often asked by youngsters to ‘chuck’ them ‘a fag’-and whole cheap cigarettes are also often called fags.
• 1908 Church Times 7 Feb. 173/1 ― He gathered into a leather pouch the remains of his cigarettes, and left the room. ‘What does he do with all those fags?’ asked Conway. 1921 Galsworthy To Let iii. viii. 284 ― The fag of Fleur’s cigarette··fell on the grass.
• 1922 Joyce Ulysses 70 ― Smoking a chewed fagbutt. 1928 Galsworthy Swan Song iv. 24 ― Cinemas, fags, and football matches-there would be no real revolution while they were on hand.
• 1942 C. Barrett On Wallaby v. 97 ― Cobbers of the men in detention had hit upon an ingenious method of smuggling fags to them.
• 1945 L. Shelly Jive Talk Dict. 24/1 ― Fag hag, girl chain smoker.
• 1959 W. Golding Free Fall ii. 49 ― There was the business of the fagcards. We all collected them.

Fag-end itself, always according to the OED, comes from an usage of fag, meaning "something that hangs loose". I read all this as suggesting that the word for cigarette came about because of the way cigarettes often hang from smokers' mouths. For a famous example, see Lucky Luke:

Solution 2:

Without any evidence, the origin of fag may be "fag end", that is the untwisted end of a rope.

-

Surprisingly, the origin of the equivalent French colloquial word, "une clope", is also unknown. Is it the same in German for "kippe" ?