• Corpus ID: 174797808

The Alon-Tarsi number of subgraphs of a planar graph

@article{Kim2019TheAN,
  title={The Alon-Tarsi number of subgraphs of a planar graph},
  author={Ringi Kim and Seog-Jin Kim and Xuding Zhu},
  journal={arXiv: Combinatorics},
  year={2019}
}
This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $\Delta(H) \le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable. 

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References

SHOWING 1-10 OF 15 REFERENCES

Defective 3-Paintability of Planar Graphs

TLDR
It is shown that every planar graph is 3-defective 3-paintable and a construction of a planargraph that is not 2-defect 3-Paintable is given.

List Improper Colourings of Planar Graphs

A graph G is m-choosable with impropriety d, or simply (m, d)*-choosable, if for every list assignment L, where [mid ]L(v)[mid ][ges ]m for every v∈V(G), there exists an L-colouring of G such that

Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency

TLDR
It is proved that, for each compact surface S, there is an integer k = k(S) such that every graph in S can be (4, k)-colored; the conjecture that 4 can be replaced by 3 in this statement is conjecture.

Planar graphs are 1-relaxed, 4-choosable

Chip Games and Paintability

TLDR
It is proved that the difference between the paint number and the choice number of a complete bipartite graph $K_{N,N}$ is $\Theta(\log \log N )$ and this result translates to the framework of on-line coloring of uniform hypergraphs.

On two generalizations of the Alon-Tarsi polynomial method

Locally planar graphs are 2-defective 4-paintable

Defective List Colorings of Planar Graphs

We combine the concepts of list colorings of graphs with the concept of defective colorings of graphs and introduce the concept of defective list colorings. We apply these concepts to vertex

Colorings and orientations of graphs

TLDR
Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques by proving that there is a legal vertex-coloring of G assigning to each vertexv a color fromS(v).

Every Planar Graph Is 5-Choosable

We prove the statement of the title, which was conjectured in 1975 by V. G. Vizing and, independently, in 1979 by P. Erdos, A. L. Rubin, and H Taylor.