A folk theorem is a mathematical fact that is generally known (at least by experts), even though its proof may never have appeared in print, or may be difficult to locate. Recently, cstheory user Eli asked for a citation to a proof of the following folk theorem.

If $latex G$ is a graph with maximum degree 3 and is a minor of $latex H$, then $latex G$ is a topological minor of $latex H$.

Wikipedia mentions this to be true, and refers to Reinhard Diestel’s excellent book on graph theory. Diestel’s book asserts its truth as well, but does not give a proof.

Graph $latex G$ is a minor of graph $latex H$, if $latex G$ can be obtained from $latex H$ by removing edges in $latex H$ and identifying the vertices that were previously the endpoints of the now-deleted edge. So, in a sense, we can say that $latex H$ “has local structures that looks like” $latex G$ if $latex G$ is a minor of $latex H$. It is sometimes possible to prove strong properties about the global structure of $latex H$ if we are guaranteed that $latex H$ has no minors of a certain type, because then no local area of $latex H$ can look a certain way, so it is possible to say something about the entire graph. The most famous example of this is Wagner’s Theorem, which states that a graph is planar iff neither $latex K_5$ nor $latex K_{3,3}$ is a minor of it.

A topological minor is a related notion. Informally, $latex G$ is a topological minor of $latex H$ if we can start with $latex G$, and transform it into a subgraph of $latex H$ by only adding vertices into edges that already exist (i.e., subdividing edges, so edge $latex (u,v)$ becomes two edges, $latex (u,x)$ and $latex (x,v)$). Figure 1 on this web page shows examples of a minor and a topological minor. (Note: that figure may be behind a paywall. If anyone knows a good diagram explaining minor vs. topological minor that is freely available, please put it in the comments.)

Fortunately for Eli, Robin Kothari and co-authors had needed this folk theorem for a paper, and they had been unable to find a proof of it either, so they reproved it themselves. Kothari’s proof (and answer to Eli’s question) appears here. It is an example of a growing number of original proofs that appear on cstheory. That proof by Kothari is definitely my “favorite answer of the week.” As Eli put it, “This is stellar.”

Filed under Answer of the Week

Tagged: graph minor, graph theory, original proof

“G is a topological minor of H if we can start with G, and transform it into H”

would it be

“G is a topological minor of H if we can start with G, and transform it into minor of H”

oh, sorry… i mean “G is a topological minor of H if we can start with G, and transform it into subgraph of H”

Yes, quite so! I will fix the post now.