The r-Dynamic Chromatic Number is Bounded in the Strong 2-Coloring Number
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915117933789184 |
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| author | Goetze, Miriam Ueckerdt, Torsten |
| author_facet | Goetze, Miriam Ueckerdt, Torsten |
| contents | A proper vertex-coloring of a graph is $r$-dynamic if the neighbors of each vertex $v$ receive at least $\min(r, \mathrm{deg}(v))$ different colors. In this note, we prove that if $G$ has a strong $2$-coloring number at most $k$, then $G$ admits an $r$-dynamic coloring with no more than $(k-1)r+1$ colors. As a consequence, for every class of graphs of bounded expansion, the $r$-dynamic chromatic number is bounded by a linear function in $r$. We give a concrete upper bound for graphs of bounded row-treewidth, which includes for example all planar graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13617 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The r-Dynamic Chromatic Number is Bounded in the Strong 2-Coloring Number Goetze, Miriam Ueckerdt, Torsten Combinatorics Discrete Mathematics A proper vertex-coloring of a graph is $r$-dynamic if the neighbors of each vertex $v$ receive at least $\min(r, \mathrm{deg}(v))$ different colors. In this note, we prove that if $G$ has a strong $2$-coloring number at most $k$, then $G$ admits an $r$-dynamic coloring with no more than $(k-1)r+1$ colors. As a consequence, for every class of graphs of bounded expansion, the $r$-dynamic chromatic number is bounded by a linear function in $r$. We give a concrete upper bound for graphs of bounded row-treewidth, which includes for example all planar graphs. |
| title | The r-Dynamic Chromatic Number is Bounded in the Strong 2-Coloring Number |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2501.13617 |