Related papers: Contractibility and the Hadwiger Conjecture

A dominating $K_t$ minor in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise disjoint non-empty connected subgraphs of $G$, such that for $1 \leq i<j\leq t$, every vertex in $T_j$ has a neighbor in $T_i$. Replacing ``every vertex in…

Hadwiger's Conjecture asserts that every $K_h$-minor-free graph is properly $(h-1)$-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with…

We prove that for any $\varepsilon>0$, for any large enough $t$, there is a graph $G$ that admits no $K_t$-minor but admits a $(\frac32-\varepsilon)t$-colouring that is "frozen" with respect to Kempe changes, i.e. any two colour classes…

Hadwiger's conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colorable. It is known that the exact version of Hadwiger's conjecture does not extend to list coloring, but it has been conjectured by Kawarabayashi and Mohar…

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…

Hadwiger's conjecture asserts that every graph with chromatic number $t$ contains a complete minor of order $t$. Given integers $n \ge 2k+1 \ge 5$, the Kneser graph $K(n, k)$ is the graph with vertices the $k$-subsets of an $n$-set such…

Given a graph $H$, let us denote by $f_\chi(H)$ and $f_\ell(H)$, respectively, the maximum chromatic number and the maximum list chromatic number of $H$-minor-free graphs. Hadwiger's famous coloring conjecture from 1943 states that…

Hadwiger conjectured in 1943 that for every integer $t \ge 1$, every graph with no $K_t$ minor is $(t-1)$-colorable. Kostochka, and independently Thomason, proved every graph with no $K_t$ minor is $O(t(\log t)^{1/2})$-colorable. Recently,…

Assume $\lambda=\{k_1,k_2, \ldots, k_q\}$ is a partition of $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a $k_\lambda$-list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be…

A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $\alpha(G)=2$ is closely related to Hadwiger's conjecture for graphs…

Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the…

Hadwiger's conjecture claims that any graph with no $K_t$ minor is $(t - 1)$-colorable. This has been proved for $t \le 6$, but remains open for $t \ge 7$. As a variant of this conjecture, graphs with no $K_t^=$ minor have been considered,…

We prove Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs, where the co-claw is the disjoint union of a triangle and a vertex, the co-gem is the disjoint union of a…

Let $G$ be a $3$-connected graph. A set $W \subset V(G)$ is called contractible if $G(W)$ is a connected graph and $G - W$ is a $2$-connected graph. In 1994, McCuaig and Ota conjectured that for any $k \in \mathbb{N}$ there exists $n \in…

We prove that every $3$-graph $H$ on $n$ vertices with minimum codegree $\delta_2(H) \geq 7n/9 + o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that $\delta_2(H) \geq 4n/5 + o(n)$…

The Hadwiger number of a graph $G$, denoted $h(G)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h(G) \ge \chi(G)$, where $\chi(G)$ denotes…

We propose local versions of Hadwiger's Conjecture, where only balls of radius $\Omega(\log(v(G)))$ around each vertex are required to be $K_{t}$-minor-free. We ask: if a graph is locally-$K_{t}$-minor-free, is it $t$-colourable? We show…

In 2014, Keevash proved the existence of $(n,q,r)$-Steiner systems (equivalently $K_q^r$-decompositions of $K_n^r$) for all large enough $n$ satisfying the necessary divisibility conditions. In 2021, Glock, K\"uhn, and Osthus proposed a…

Motivated by the famous Hadwiger's Conjecture, we study the properties of $8$-contraction-critical graphs with no $K_7$ minor; we prove that every $8$-contraction-critical graph with no $K_7$ minor has at most one vertex of degree $8$,…

Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable…