Related papers: Excluding $K_{2,t}$ as a fat minor

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. While Hadwiger's conjecture does not hold for list-coloring, the linear weakening is conjectured to be true. In the 1980s, Kostochka…

The celebrated Hadwiger's conjecture states that if a graph contains no $K_{t+1}$ minor then it is $t$-colourable. If true, it would in particular imply that every $n$-vertex $K_{t+1}$-minor-free graph has an independent set of size at…

We show that any self-complementary graph with $n$ vertices contains a $K_{\lfloor \frac{n+1}{2}\rfloor}$ minor. We derive topological properties of self-complementary graphs.

Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper $(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via improper colourings. We prove that every $K_t$-minor-free graph is $(2t-2)$-colourable with…

For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[…

We prove that for every integer $t\ge 1$ there exists a constant $c_t$ such that for every $K_t$-minor-free graph $G$, and every set $S$ of balls in $G$, the minimum size of a set of vertices of $G$ intersecting all the balls of $S$ is at…

In this paper we use theoretical and computational tools to continue our investigation of $K_2$-hamiltonian graphs, that is, graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, and their interplay with…

We show that graphs excluding $K_{2,t}$ as a minor admit a $f(t)$-round $50$-approximation deterministic distributed algorithm for Minimum Dominating Set. The result extends to Minimum Vertex Cover. Though fast and approximate distributed…

A graph $H$ is an \emph{induced minor} of a graph $G$ if $H$ can be obtained from $G$ by a sequence of edge contractions and vertex deletions. Otherwise, $G$ is \emph{$H$-induced minor-free}. In this paper, we provide a different proof of…

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…

It is an easy observation that if a graph~$G$ admits a path-decomposition whose parts have small radius, then $G$ contains no large subdivision of $K_{1,3}$ or $K^3$ as a (quasi-)geodesic subgraph. We show that these are in fact the only…

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…

In 1943, Hadwiger conjectured that every $K_t$-minor-free graph is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$…

An \emph{$(n,k,t)$-graph} is a graph on $n$ vertices in which every set of $k$ vertices contains a clique on $t$ vertices. Tur\'an's Theorem, rephrased in terms of graph complements, states that the unique minimum $(n,k,2)$-graph is an…

We investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t-1 vertices is not t-1 colorable, so is conjectured to have a $K_t$ minor. There is a strengthening of Hadwiger's conjecture in this…

The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if it does not have either $K_{3,3}$ or $K_5$ as a minor. Using this Wagner obtained a precise description of all graphs with no $K_{3,3}$ minor and all graphs with no…

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$.

Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log…

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log…

The spread of a graph $G$ is the difference between the largest and smallest eigenvalues of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{2,t}$-minor. We show that for any $t\geq 2$, there…