Search Results

I'll just remark that the fact that every surface has a finite number of excluded minors (and also topological minors) does not require the full strength of the Graph Minors Theorem. Indeed, the proo …

As others have mentioned, the answer to your title question is strictly speaking no. With regards to your other questions, it has been proven that it is undecidable to compute the excluded minors for …

More generally, it is a classic result (independently due to Kostochka and Thomason) that minimum degree $(\alpha+o(1))n \sqrt{\log n}$ suffices to force a $K_n$ minor, where $\alpha$ is an explicit c …

Yes. This result is contained in my PhD thesis, which is available here (see Theorem 1.1.10). We prove that for any finite abelian group $\Gamma$ and fixed $\Gamma$-labeled graph $H$, there is a pol …

Yes. This is one of the results of the Matroid Minors Project of Geelen, Gerards and Whittle as part of their proof of Rota's Conjecture. In fact, they prove that for any finite abelian group $\Gamm …

Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $K_5$ minor can be built from $0$-, $1$-, $2$-, and $3$-sums from planar graphs and a fixed $8$ …

For your first question, Mader proved that all $K_6$-minor-free graphs (which includes all linklessly embeddable graphs) on $n$ vertices have at most $4n-10$ edges (thanks to David Eppstein for the re …

There is no known necessary and sufficient condition like in Menger's theorem. However, there is a polynomial-time algorithm that decides if the paths exist. This is one of the main results of the Gr …

Here is another answer that I just recently learned about that I find quite shocking. A graph is called $Y \Delta Y$ reducible if it can be reduced to isolated vertices by applying any of the followi …

The property that you are describing is called $2$-linked. More generally, we say that a graph is $k$-linked if it has at least $2k$ vertices and for all distinct vertices $s_1, \dots s_k, t_1, \dots …

This is false by classic results of Kostochka and Thomason. Indeed, the claim is false even if you replace 'minimum degree $t$' with '$t$-connected'. That is, if you define $\nu(t)$ to be the smalle …

Yes, in my PhD thesis, we prove the stronger result that for any fixed signed graph $(G, \Sigma)$, there is a polynomial-time algorithm to test if an input signed graph contains a $(G, \Sigma)$-minor. …

As far as I understand, the purported proof does not give an algorithm that given a finite field $\mathbb{F}$, computes the excluded minors for $\mathbb{F}$-representability. This is because it relie …

No, there is no such graph. Suppose $\eta(G)=n$. Let $T_1, \dots, T_n$ be a collection of vertex disjoint trees in $G$ such that for all distinct $i,j \in [n]$, there is an edge $e(ij) \in E(G)$ betw …

No, your inequality does not hold. You are off by a constant factor. Probably the easiest way to see this is to consider the dual notion of a tangle, which I will define now. A separation in a graph …