The Alon-Tarsi number of subgraphs of a planar graph

Abstract

This paper constructs a planar graph G 1 such that for any subgraph H of G 1 with maximum degree ∆(H) ≤ 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.

Schnyder Woods and Alon-Tarsi Number of Planar Graphs

“…We illustrate how the basic results on the realizers of planar graphs can be used to derive an alternative proof of the following strengthening of one of the results from [7]. (The actual bound from [7] was for the Alon-Tarsi number.) Theorem 13 (Kim, Kim and Zhu [7]).…”

Schnyder woods and Alon-Tarsi number of planar graphs

“…We illustrate how the basic results on the realizers of planar graphs can be used to derive an alternative proof of the following strengthening of one of the results from [7]. (The actual bound from [7] was for the Alon-Tarsi number.) Theorem 2.11 ([7]).…”

The Alon-Tarsi number of $K_5$-minor-free graphs

“…Since if it was true, this implies that every planar graph is 2-defective 3-choosable. However,this is not true and it was shown in [8] that there exists a planar graph G such that for any subgraph of H of G with maximum degree at most 3, G − E(H) is not 3-choosable. On the other hand, the following was also proved in the same paper.…”

“…We say a plane graph G is a near triangulation if each internal face in G is triangular. In the papers [5], [8] and [15], the following are shown respectively.…”