# Boxicity, Poset Dimension, and Excluded Minors

@article{Esperet2018BoxicityPD, title={Boxicity, Poset Dimension, and Excluded Minors}, author={Louis Esperet and Veit Wiechert}, journal={Electron. J. Comb.}, year={2018}, volume={25}, pages={4} }

In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.

## 6 Citations

### Local Boxicity, Local Dimension, and Maximum Degree

- MathematicsArXiv
- 2018

An `almost linear' upper bound is given for both the parameters in terms of the maximum degree of a graph, namely `local boxicity' and `local dimension', and two other upper bounds for the local boxicity are proved.

### Better bounds for poset dimension and boxicity

- Mathematics, Computer ScienceTransactions of the American Mathematical Society
- 2019

It is proved that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta\log^{1+o(1)} \Delta$, and is within a $\log^{o( 1)}\Delta$ factor of optimal.

### Shallow Minors, Graph Products and Beyond Planar Graphs

- Mathematics
- 2021

The planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with…

### Odd colourings, conflict-free colourings and strong colouring numbers

- Mathematics
- 2022

The odd chromatic number and the conflict-free chromatic number are new graph parameters introduced by Petruševski and Škrekovski [2021] and Fabrici, Lužar, Rindošová and Soták [2022] respectively.…

### Bounding threshold dimension: realizing graphic Boolean functions as the AND of majority functions

- Mathematics
- 2022

A graph G on n vertices is a threshold graph if there exist real numbers a1, a2, . . . , an and b such that the zero-one solutions of the linear inequality n ∑ i=1 aixi ≤ b are the characteristic…

### Bounding Threshold Dimension: Realizing Graphic Boolean Functions as the AND of Majority Gates

- MathematicsWG
- 2022

A graph G on n vertices is a threshold graph if there exist real numbers a1, a2, . . . , an and b such that the zero-one solutions of the linear inequality n ∑ i=1 aixi ≤ b are the characteristic…

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