Which graphic file format is best to use in printing technology ?

Well .ai format will be better for the print, following are the reasons why: 1.
As its a vector format which can be scaled at any level without losing quality.
2.
An editable format.
3.
This can be exported easily in different formats as per the needs.
4.
Its a source format which gives more flexibility in editing & printing comparing than PDF.

There are a lot of smaller points we should consider before selecting any format for sending artwork to printing press.
1.
Color separation options should be there.
2.
Any option to retain Fonts' conditions should also be there e.g.
"Convert to Curves option in Corel Draw".
3.
You should use "CMYK" Color Palettes.
If you'll use the "RGB" scheme the printed output would not match your expected colors.
Luckily Corel Draw has all these features available.
It can Convert the Fonts in to Curves so these can open and retain visibility everywhere, it can separate the colors and have wonderful CMYK color palettes available.
In last I would suggest the "CDR" corel draw native file format for sharing files with your printing guys.

pdf or eps 

to color seperation offset printing

 

Tiffe 

to plotor printing ( inndoor or out door printing )

 

the encapsulated post script

Is the best of format for out put printing and seperation color is original sowftware format from your design for example.
, if you maked your design in adobe illustrator is better you embeded your photo and curve your font on design then make seperation color and check it in adobe acrobat with pdf format .

PDF - that way your design and fonts are preserved and can not be altered.

I think, CDR format file (Corel Draw) is good to print your graphic

pdf or ai

The best solution is to call the printing house with which you want to cooperate and ask about the guidelines. By default, a PDF is sent (raster graphics should have 300dpi resolution for small printouts, the bigger sizes can have smaller resolition for example 150dpi - the size of vector graphics doesn't matter - but please remember that if the lines in the graphics are too thin, the printer don't prints them - I encorage to consultation with the printing house ) 

If you have any question, don't hesitate to contact me. Good luck :)

.PDF files are the best. Specially if they are editable in Adobe illustrator. Since they can be previewed on any device by any non-designer person

*.tif and format cmyk