What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found.


A good example is Lovász's proof of [Kneser's conjecture](https://en.wikipedia.org/wiki/Kneser_graph).



Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?
The question asks for *concepts*, not applications, so in a sense the example given in the OP isn't one. 


Here are five quick examples: 


(0) One could argue that **girth** is a transferral of the concept **systole** from metric-topology, though this is an ahistorical argumentation: the two concepts arose independently in their respective fields. 


(1) The *concept* of associated abstract polyhedral/simplicial **complexes**. This is the central concept in both Lovász's proof of Kneser's conjecture, and in the proof in [Babson--Kozlov, *Proof of the Lovász conjecture*, Annals of Mathematics (20 165 (2007) 965-1007]


(2) The *concept* of **topological connectivity** (i.e. (1+) smallest dimension such that there exists a non-nullhomotopic continuous map from a correspondingly-dimensional sphere to the geometric realization of the complex (this is strongly related to *what the opening poster already mentioned*, yet it was not explicitly mentioned yet.


(3) The similar *concepts* of **Whitney complex** and **barycentric subdivision** (see recent work of Oliver Knill). 


(4) The *concept* of Freudenthal-**compactification** (see 'ends of graphs'), an important tool to enlarge the language with which to speak about infinite graphs. 


