The List Hadwiger Conjecture asserts that every Kt-minor-free graph is t-choosable. We disprove this conjecture by constructing a K3t+2-minor-free graph that is not 4tchoosable for every integer t ≥ 1.

We prove that a 4-connected K 4;4-minor free graph on n vertices has at most 4n ? 8 edges and we use this result to show that every K 4;4-minor free graph has vertex-arboricity at most 4. This improves the case (n; m) = (7; 3) of the following conjecture of Woodall: the vertexset of a graph without a K n-minor and without a K b n+1 2 c;d n+1 2 e-minor can be partitioned in n ? m + 1 subgraphs w...

The List Hadwiger Conjecture asserts that every Kt-minor-free graph is tchoosable. We disprove this conjecture by constructing a K3t+2-minor-free graph that is not 4t-choosable for every integer t ≥ 1.

Hadwiger’s Conjecture [7] states that every Kt+1-minor-free graph is t-colourable. It is widely considered to be one of the most important conjectures in graph theory; see [21] for a survey. If every Kt+1-minor-free graph has minimum degree at most δ, then every Kt+1minor-free graph is (δ+1)-colourable by a minimum-degree-greedy algorithm. The purpose of this note is to prove a slightly better ...

A graph H is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor-free. Robin Thomas showed that K4-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without K4 and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 – 247, 1985]. We provide a dichotomy th...

Tutte [W.T. Tutte, On the algebraic theory of graph colorings, J. Combin. Theory 1 (1966) 15–20] conjectured that every bridgeless Petersen-minor free graph admits a nowhere-zero 4-flow. Let (P10)μ̄ be the graph obtained from the Petersen graph by contracting μ edges from a perfect matching. In this paper we prove that every bridgeless (P10)3̄-minor free graph admits a nowhere-zero 4-flow. c © 20...

Local certification consists in assigning labels to the nodes of a network certify that some given property is satisfied, such way can be checked locally. In last few years, graph classes received considerable attention. The goal $G$ belongs class~$\mathcal{G}$. Such certifications with size $O(\log n)$ (where $n$ network) exist for trees, planar graphs and embedded on surfaces. Feuilloley et a...

For each proper subgraph H of K5, we determine all pairs (k, d) such that every H-minor-free graph is (k, d)-choosable or (k, d)-choosable. The main structural lemma is that the only 3-connected (K5 − e)-minor-free graphs are wheels, the triangular prism, and K3,3; this is used to prove that every (K5 − e)-minor-free graph is 4-choosable and (3, 1)-choosable.