Management competence is one of the main skills that the labor market expects from everyone graduating from tertiary education.
Especially, I have some doubt about the relative clause. Is it acceptable to use “from everyone graduating” instead of “from everyone who is graduating”?
I’m also confused with the expression “graduating from tertiary education”. Is it possible to use the expression “to graduate from education” and not "to graduate from an academic institution (e.g. college, university)?
I am 99.99% certain that Americans would never say something like “tertiary education.” After a few moments’ thought, they might remember that “high school” is secondary education, so they might realize that you are referring to the university (or “college,” as many Americans call it).
I am 99.99% certain that “to graduate from education” is NOT acceptable.
Remember, please, that a person IS GRADUATED from a SCHOOL.
But in 2014, most people no longer use the “is”: I graduated from the University of Southern California in 1959. (That’s true. I am so old!)
Therefore, you could say something like:
“Management competence is one of the main skills that the labor market expects from every college graduate.”
The sentence is okay. In the relative clause ‘who is’ is assumed and accepted, as is 'graduating from (an institution providing) tertiary education.
It’s not ideal as a sentence in a very formal situation, but it is passable in standard use.
EDIT:
I posted at the same time as James, and would agree with the point he makes regarding the term ‘tertiary education’. Most people would not use it. However, in this case, I presume it relates to the standard ‘business speak’ type term for the company responsible for the document. That is, the company and their customer base would use the term because they work within and around the field of tertiary education.