I'm reading Van Lint and Wilson: A Course in Combinatorics.
There is one part of the book where he defines a finite graph:
A graph is ﬁnite when both E(G) and V (G) are ﬁnite sets.
Theorem 1.1. A ﬁnite graph G has an even number of vertices with odd valency.
Reading it further, there is:
Theorem 1.2. A ﬁnite graph G with no isolated vertices (but possibly with multiple edges) is Eulerian if and only if it is connected and every vertex has even degree.
It was first said that a finite graph has an odd valency at each vertex, now a finite graph with every vertex with even degree is presented. How should I proceed? Should I drop the condition given in the first theorem (odd valency)? Does Eulerian graph mean a finite graph without this condition? I believe it is, but I want to be sure.