Meaning of "Up to isomorphism"

Question

I saw in my graph theory notes this statement "Up to isomorphism, there is one and only one $K_4$". What does the phrase "up to isomorphism" mean?

Answer 1

My answer here is just going to expand on Lee Mosher's comment for the sake of answering this question.

The phrase "up to isomorphism" means that we are defining two things to be equivalent if and only if they are isomorphic. The reason we need to say this is that in graph theory there is no obvious definition of two graphs being equivalent. For example, look a the following three graphs:

These graph certainly look distinct, and if you were to define graph equivalency based on how the graph was drawn/embedded (kinda like we use congruency as an equivalence on triangles), then these graphs would be distinct. But "up to isomorphism", these these graphs are actually the same graph ($K_4$), because each one is isomorphic to the other two. To rephrase that statement in your notes, "No matter how you draw $K_4$, it is still $K_4$".

Similarly if we look at the two graphs below:

They are both clearly $K_{3,3}$, so "up to isomorphism" they are the same graph. But you might not want to consider them equivalent because the vertices are labelled differently (or just because they are different colors).