• Corpus ID: 219965889

# Further progress towards Hadwiger's conjecture

@article{Postle2020FurtherPT,
author={Luke Postle},
journal={ArXiv},
year={2020},
volume={abs/2006.11798}
}
• L. Postle
• Published 21 June 2020
• Computer Science, Mathematics
• ArXiv

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

• Mathematics
ArXiv
• 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

• Mathematics
Comb. 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

• R. Steiner
• Mathematics
Combinatorics, 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

### Clique immersions and independence number

• Mathematics
European Journal of Combinatorics
• 2022

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

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

SHOWING 1-10 OF 33 REFERENCES

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

• L. Postle
• Computer Science, Mathematics
ArXiv
• 2020

### Fractional Colouring and Hadwiger's Conjecture

• Mathematics
J. 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.

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