You may have heard that "TG&Y" stood for "Turtles, Girdles & Yo-yos." Seriously, the chain's name was an amalgamation of the surnames of its three founders. Left-to-right, they were Rawdon Tomlinson (1885-1948), Enoch Gosselin (1901-1977) and Raymond Young (1904-2002). BTW, the order of the initials coincided with the ages of three founders, with Tomlinson being the oldest. 

Photos 1 & 2 from https://www.findagrave.com

Photo 3 from https://oklahomahof.com / Oklahoma Hall of Fame

A vintage newspaper ad from 1936 announces the grand opening of the very first TG&Y dime store, which was located in Norman, Oklahoma.

Advert from The TG&Y Stores Company

TG&Y opened -in Midwest City, Oklahoma's ATKINSON PLAZA- in July 1943. This complex was one of the first suburban shopping centers in the American heartland.

Photo from White Chipmunk's Photostream / Atkinson Heritage Center

The first TG&Y units were not promoted with a bona fide logo. This basic heading would suffice until a graphically-designed trademark was created.

Graphic from the T G & Y Stores Company 

Said trademark first appeared in print ads in 1937. 

Graphic from the T G & Y Stores Company 

This logo was in use in 1945.

Graphic from the T G & Y Stores Company 

The quintessential TG&Y trademark was introduced in 1949.

Graphic from the T G & Y Stores Company 

There would be many artistic variations on the basic 1949 trademark. This design was in use in 1968 and continued to be seen in print ads through the mid-1980s.

Graphic from the T G & Y Stores Company 

By 1995, the McCrory Corporation had bought out several once-prosperous 5 & 10 operations, including TG&Y. Individual store chains were operated under their original nameplates, but a combo-trademark graphic was used in print ads. The final TG&Y variety stores were shuttered in 2001.
Graphic from the McCrory Corporation