Are there any graph theory results which clearly fail for non-simple graphs? Many times when I do graph theory, I might write something like "assume for simplicity that $G$ is a simple graph", thinking that this doesn't really pose a major problem and some elementary modifications to the argument could solve the general case.
Are there (non-trivial, non-facetious) results in graph theory for which this isn't the case? (The theorem "every graph is a simple graph" is a facetious example).
 A: Out of the topics typically covered in an introductory graph theory course:

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*Many degree conditions for graphs having a certain property are false for multigraphs. For example, we cannot say that an $n$-vertex multigraph with minimum degree at least $\frac{n-1}{2}$ is connected, or that with minimum degree $\frac n2$ it is Hamiltonian. (There's lots of degree conditions for Hamiltonicity, and all of them require the graph to be simple.) A notable exception is results about matchings in bipartite graphs (like Hall's theorem or König's theorem), which still hold for multigraphs.

*Similarly, results about the maximum number of edges in a graph without some property (of which the simplest is probably Mantel's theorem: an $n$-vertex triangle-free graph has at most $\frac{n^2}{4}$ edges) do not hold for multigraphs.

*The problem of realizing a degree sequence is more or less trivial for multigraphs, but more complicated for simple graphs.

*Most results about planar graphs continue to hold for multigraphs, but the edge upper bound $m \le 3n-6$ requires the graph to be simple, and some fancier results also require faces to have length at least $3$. Also, the connectivity of a planar graph $G$ is related to the presence of loops and parallel edges in the dual of a plane embedding of $G$.

It is also common to forbid loops (but not parallel edges) when studying:

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*Connectivity of graphs, where loops don't help us connect different vertices but may mess with the degree conditions.

*Graph coloring, where a single loop makes a graph impossible to properly color. (Parallel edges also do nothing here, but in particular they do no harm.)

