Definition
Countable graph
A graph whose vertex set is finite or countably infinite.
A countable graph is a graph whose vertex set is countable, meaning finite or countably infinite. Its vertices can therefore be processed in a finite list or a sequence containing each vertex exactly once.
Sequential constructions
For a simple graph on a countable vertex set, the edge set is countable too, since edges are unordered pairs of distinct vertices. Countability supports sequential choices such as greedy coloring. It does not bound the degree: a vertex may have infinitely many neighbors.