“Cry calf rope” (or “Holler calf rope” or “Yell calf rope”)

“Cry calf rope” (or “holler calf rope” or “yell calf rope”) means the same as “cry uncle”—an admission of defeat. It comes from the phrase that if you give a calf enough rope, he’ll hang himself. The “calf rope” expression has been widely used in Texas for well over a century.
 
 
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Fisher’s River (North Carolina) Scenes and Characters
by Hardin E. Taliaferro
New York: Harper & Brothers
1859
Pg. 87:
“So we moseyed into ‘um yearly in the mornin’, and ‘bout twelve o’clock he called fur the calf-rope. I’d beat him all holler.”
   
Making ofAmerica
December 1877, Scribner’s Monthly, “Roxy” by Edward Eggleston, pg. 195:
..finding that his antagonist had entirely stopped respiration, was forced to “howllow calf-rope,” that is, to signify by gestures that he was beaten.
 
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Cum Along Thru’ Texas
by Helen B. Odom
Kessinger Publishing
2005
original copyright 1936
Pg. 122:
Holler Calf Rope…Tain’t like ye hollers erbout tyin’ up a calf with th’ rope. It means a feller whot kain’t stan’ the’ gaff, then hollers…an’ yew giv’s th’ sucker plenty uv rope sos he kin choke hisse’f.
   
Google Books
To Tempt a Texan
by Georgina Gentry
Zebra Books
2005
Pg. 124:
“Anyway, knowing her like I do, I reckon you’ll yell ‘calf rope’ before she does.” 
 
Lone Star BeerGil King 9 July 2005
A Great Beer , especially Lone Star Light. Good enough to cure rabbies, babies, acorns, corns, hangovers and hangnails !!! Keep em comin till I yell ” Calf Rope ”
 
Texas Tall Tales
Located in the Panhandle city of Amarillo, Texas, The Big Texan Steak Ranch offers a challenging dinner special.  The meal consists of a 72 ounce top sirloin steak, salad, baked potato, dinner roll and shrimp cocktail. If you don’t yell “calf rope” and are able to finish it within the given hour time limit, the entire meal is free. 
 
but that’s just my opinion
Sunday, July 09, 2006
If the Palestinians support hostage taking, then so be it. Bomb their ass till they yell calf-rope (old Texas phrase meaning I give up!)
 
Do You Speak American - Track That Word!
(Source: Dictionary of American Regional English)
Calf-rope, holler (1878)
Esp in children’s games; to give in, surrender; to capitulate.
         
6 February 1878, Daily Nevada State Journal (Reno, NV), pg. 2, col. 2: 
Of Kearney and his retainers it would seem the thing to let them whip themselves to death, or as the homely adage has it, “Give a calf rope and he will break his own neck.”
 
1 October 1900, Galveston (TX) Daily News, pg. 4, col. 4:
Some of them would no doubt be glad to cry “calf rope” and play quits.
 
24 February 1915, Wichita Falls (TX) Daily Times, pg. 9, col. 1 ad:
Some folks sit in the shade of indolence and expect luck to come along, lambast them in the back until they holler “Calf Rope,” then drag them up the hill of success while acclaiming friends stand by shouting glory.
 
16 December 1938, Austin (TX) Statesman, pg. 17, col. 6:
Coach Weldon Chapmans mighty men of the South Plains were heavy favorites to defeat the rough and ready though light and outmanned Masons who have yet to yell “calf rope” no matter what the odds might be.
 
11 January 1942, Galveston (TX) Daily News, pg. 4, col. 7:
If Boreas is waiting for someone to yell “calf rope,” Slanis in the absence of the city’s mayor, hereby so yells.