Weak Degeneracy of Planar Graphs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910987755454464 |
|---|---|
| author | Bernshteyn, Anton Lee, Eugene Smith-Roberge, Evelyne |
| author_facet | Bernshteyn, Anton Lee, Eugene Smith-Roberge, Evelyne |
| contents | The weak degeneracy of a graph $G$ is a numerical parameter that was recently introduced by the first two authors with the aim of understanding the power of greedy algorithms for graph coloring. Every $d$-degenerate graph is weakly $d$-degenerate, but the converse is not true in general (for example, all connected $d$-regular graphs except cycles and cliques are weakly $(d-1)$-degenerate). If $G$ is weakly $d$-degenerate, then the list-chromatic number of $G$ is at most $d+1$, and the same upper bound holds for various other parameters such as the DP-chromatic number and the paint number. Here we rectify a mistake in a paper of the first two authors and give a correct proof that planar graphs are weakly $4$-degenerate, strengthening the famous result of Thomassen that planar graphs are $5$-list-colorable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02792 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak Degeneracy of Planar Graphs Bernshteyn, Anton Lee, Eugene Smith-Roberge, Evelyne Combinatorics The weak degeneracy of a graph $G$ is a numerical parameter that was recently introduced by the first two authors with the aim of understanding the power of greedy algorithms for graph coloring. Every $d$-degenerate graph is weakly $d$-degenerate, but the converse is not true in general (for example, all connected $d$-regular graphs except cycles and cliques are weakly $(d-1)$-degenerate). If $G$ is weakly $d$-degenerate, then the list-chromatic number of $G$ is at most $d+1$, and the same upper bound holds for various other parameters such as the DP-chromatic number and the paint number. Here we rectify a mistake in a paper of the first two authors and give a correct proof that planar graphs are weakly $4$-degenerate, strengthening the famous result of Thomassen that planar graphs are $5$-list-colorable. |
| title | Weak Degeneracy of Planar Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.02792 |