Related papers: K_6 minors in 6-connected graphs of bounded tree-w…

Jorgensen conjectured that every 6-connected graph with no K_6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.

We prove that every 6-connected graph of girth $\geq 6$ has a $K_6$-minor and thus settle the Jorgensen conjecture for graphs of girth $ \geq 6$. Relaxing the assumption on the girth, we prove that every 6-connected $n$-vertex graph of size…

We show that all sufficiently large (2k+3)-connected graphs of bounded tree-width are k-linked. Thomassen has conjectured that all sufficiently large (2k+2)-connected graphs are k-linked.

The purpose of this paper is to characterize graphs that do not have a large $K_{2,n}$-minor. As corollaries, it is proved that, for any given positive integer $n$, every sufficiently large 3-connected graph with minimum degree at least…

A graph is apex if it becomes planar after the deletion of one vertex. The family of apex graphs is closed under taking minors, so it is characterized by a finite set of forbidden minors. Determining the finite set of forbidden minors for…

We prove that every simple graph of order 12 which has minimum degree 6 contains a K_6 minor.

A complete structural characterization of graphs with no $K_{3,4}$ minor is obtained, and the following consequences are established. Every $4$-connected non-planar graph with at least seven vertices and minimum degree at least five…

For each integer $n \geq 3$, the wheel graph $W_n$ is defined as the graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ can be uniquely obtained from the Petersen graph by contracting…

Let $P$ be a graph with a vertex $v$ such that $P\backslash v$ is a forest, and let $Q$ be an outerplanar graph. We prove that there exists a number $p=p(P,Q)$ such that every 2-connected graph of path-width at least $p$ has a minor…

For each integer $n\geq 3$, let $W_n$ denote the wheel graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ is obtained from the Petersen graph by contracting three edges incident with…

A theorem of Mader shows that every graph with average degree at least eight has a $K_6$ minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do…

We prove that every connected graph $G$ with $m$ edges contains a set $X$ of at most $\frac{3}{16}(m + 1)$ vertices such that $G-X$ has no $K_4$ minor, or equivalently, has treewidth at most $2$. This bound is best possible. Connectivity is…

It was proved by Huynh, Mohar, \v{S}\'amal, Thomassen and Wood in 2021 that any countable graph containing every countable planar graph as a subgraph has an infinite clique minor. We prove a finite, quantitative version of this result: for…

A classical result of Robertson and Seymour (1986) states that the treewidth of a graph is linearly tied to its separation number: the smallest integer $k$ such that, for every weighting of the vertices, the graph admits a balanced…

It is proved that every graph embedded on a (non-spherical) surface with non-separating face-width at least $7$ contains a minor isomorphic to $K_6$. It is also shown that face-width four yields the same conclusion for graphs on the…

Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this paper, we give a…

Tutte showed that $4$-connected planar graphs are Hamiltonian, but it is well known that $3$-connected planar graphs need not be Hamiltonian. We show that $K_{2,5}$-minor-free $3$-connected planar graphs are Hamiltonian. This does not…

Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every…

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown…

In 2006 Bar{\'a}t and Thomassen conjectured that every planar $4$-edge-connected $4$-regular simple graph of size divisible by three admits a claw-decomposition. Later, Lai (2007) disproved this conjecture by a family of planar graphs with…