# Further progress towards Hadwiger's conjecture

@article{Postle2020FurtherPT, title={Further progress towards Hadwiger's conjecture}, author={Luke Postle}, journal={ArXiv}, year={2020}, volume={abs/2006.11798} }

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 $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt…

## 15 Citations

### Further Progress towards the List and Odd Versions of Hadwiger's Conjecture.

- Mathematics, Computer Science
- 2020

This paper extends the work to the list and odd generalizations of Hadwiger's conjecture and shows that every graph with no K_t minor is O(t (\log t)^{\beta})-colorable for every $\beta > 0$.

### An even better Density Increment Theorem and its application to Hadwiger's Conjecture

- Computer Science, MathematicsArXiv
- 2020

Every graph with no $K_t$ minor is $O(t (\log \log t)^{6})$-colorable, making the first improvement on the order of magnitude of the O(t\sqrt{\log t})$ bound.

### Reducing Linear Hadwiger's Conjecture to Coloring Small Graphs

- MathematicsArXiv
- 2021

Every graph with no Kt minor is O(t log log t)-colorable and Linear Hadwiger’s Conjecture reduces to proving it for small graphs, and it is proved that Kt-minor-free graphs with clique number at most √ log t/(log log t)2 are O( t)- colorable.

### Strong complete minors in digraphs

- MathematicsComb. Probab. Comput.
- 2022

Kostochka and Thomason independently showed that any graph with average degree
$\Omega(r\sqrt{\log r})$
contains a
$K_r$
minor. In particular, any graph with chromatic number…

### Improved bound for improper colourings of graphs with no odd clique minor

- MathematicsCombinatorics, Probability and Computing
- 2022

Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd
$K_t$
-minor is properly
$(t-1)$
-colourable. This is known as the…

### Asymptotic Equivalence of Hadwiger's Conjecture and its Odd Minor-Variant

- MathematicsJ. Comb. Theory, Ser. B
- 2022

### Clique immersion in graphs without a fixed bipartite graph

- MathematicsJ. Comb. Theory, Ser. B
- 2022

### Improved lower bound for the list chromatic number of graphs with no $K_t$ minor

- Mathematics
- 2021

Hadwiger’s conjecture asserts that every graph without a Kt-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…

## References

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### Halfway to Hadwiger's Conjecture

- MathematicsArXiv
- 2019

Improving on the second part of their argument, Norin and Song prove that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1}{4}$.

### An even better Density Increment Theorem and its application to Hadwiger's Conjecture

- Computer Science, MathematicsArXiv
- 2020

Every graph with no $K_t$ minor is $O(t (\log \log t)^{6})$-colorable, making the first improvement on the order of magnitude of the O(t\sqrt{\log t})$ bound.

### Breaking the degeneracy barrier for coloring graphs with no $K_t$ minor

- Mathematics
- 2019

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\geq 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$…

### A New Upper Bound on the Chromatic Number of Graphs with No Odd Kt Minor

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- 2022

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…

### Subgraphs of large connectivity and chromatic number

- MathematicsBulletin of the London Mathematical Society
- 2022

Resolving a problem raised by Norin in 2020, we show that for each k∈N$k \in \mathbb {N}$ , the minimal f(k)∈N$f(k) \in \mathbb {N}$ with the property that every graph G$G$ with chromatic number at…

### On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture

- MathematicsJ. Comb. Theory, Ser. B
- 2007

### Fractional Colouring and Hadwiger's Conjecture

- MathematicsJ. Comb. Theory, Ser. B
- 1998

It is proved that the “fractional chromatic number” of G is at most 2p; that is, it is possible to assign a rationalq(S)?0 to every stable set S?V(G) so that ?S?vq (S)=1 for every vertexv, and ?Sq( S)?2p.

### Hadwiger's Conjecture

- MathematicsOpen Problems in Mathematics
- 2016

This is a survey of Hadwiger’s conjecture from 1943, that for all t ≥ 0, every graph either can be t-coloured, or has a subgraph that can be contracted to the complete graph on t + 1 vertices. This…

### Some Recent Progress and Applications in Graph Minor Theory

- MathematicsGraphs Comb.
- 2007

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough'' structure of graphs excluding a fixed minor. This result was used to prove…