The AlonTarsi 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 $\Delta(H) \le 3$, $G_1E(H)$ is not $3$choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2E(F)$ contains a copy of $K_4$ and hence $G_2E(F)$ is not $3$colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the AlonTarsi number of $G  E(F)$ is at most $3$, and hence $G  E(F)$ is 3paintable and 3choosable.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.01506
 Bibcode:
 2019arXiv190601506K
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 12 pages, 5 figures