Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable
Abstract
Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{4,5\}$-cycles and $d^{\Delta}\ge d_0$ are $3$-choosable and planar graphs without $\{4,5,6\}$-cycles and $d^{\Delta}\ge d_1$ are $3$-choosable, where $d^{\Delta}$ is the smallest distance between triangles. They showed that $2\le d_0\le 4$ and $d_1\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\{4,5\}$-cycles and $d^{\Delta}\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\{4,5,6\}$-cycles and $d^{\Delta}\ge 2$ are DP-$3$-colorable. DP-coloring is a generalization of list-coloring, thus as a corollary, $d_0\le 3$ and $d_1\le 2$. We actually prove stronger statements that each pre-coloring on some cycles can be extended to the whole graph.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2018
- DOI:
- 10.48550/arXiv.1809.00925
- arXiv:
- arXiv:1809.00925
- Bibcode:
- 2018arXiv180900925Y
- Keywords:
-
- Mathematics - Combinatorics
- E-Print:
- 14 pages. This is an updated version of a submission. In this version, Theorem 1.3 is stronger: instead of $d_0\ge 4$ in the submission, we have $d_0\ge 3$ in this version