Related papers: K_6 minors in large 6-connected graphs

We prove that every sufficiently big 6-connected graph of bounded tree-width either has a K_6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently…

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…

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 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 $n\geq 3$, 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…

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…

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…

Let $K_7^{\vee}$ denote the graph obtained from the complete graph on seven vertices by deleting two edges with a common end. Motivated by Hadwiger's conjecture, we prove that every graph with no $K_7^{\vee}$-minor is $6$-colorable.

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

For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from the complete graph $K_t$ by removing $s$ edges. A graph $G$ has no $\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\in…

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…

Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no $K_{3,3}$ minor, and consequently for planar graphs.

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the…

Hadwiger's Conjecture from 1943 states that every graph with no $K_{t}$ minor is $(t-1)$-colorable; it remains wide open for all $t\ge 7$. For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from…

Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in…

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…

We prove that every 3-regular graph with no circuit of length less than six has a subgraph isomorphic to a subdivision of the Petersen graph.

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…

As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every…

We provide a short proof that a 5-connected nonplanar apex graph contains a subdivided $K_{_5}$ or a $K^-_{_4}$ (= $K_{_4}$ with a single edge removed) as a subgraph. Together with a recent result of Ma and Yu that {\sl every nonplanar…