Majority additive coloring and the maximum degree
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914168699879424 |
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| author | Brause, Christoph Rautenbach, Dieter Schwartze, Laurin |
| author_facet | Brause, Christoph Rautenbach, Dieter Schwartze, Laurin |
| contents | Kamyczura introduced the notion of a majority additive $k$-coloring of a graph $G$ as a function $c: V(G) \to \{1,2,\ldots,k\}$ such that $$\left|\left\{u \in N_G(v):\sum_{w \in N_G(u)} c(w) = s \right\}\right|\leq \max\left\{1,\frac{d_G(v)}{2}\right\}$$ for every vertex $v$ of $G$ and every positive integer $s$. We show that every graph $G$ of maximum degree $Δ$ admitting a majority additive coloring has a majority additive $\mathcal{O}\left(Δ^2\right)$-coloring. Under additional restrictions we improve this to sublinear in $Δ$. We show that determining whether a majority additive $k$-coloring exists for a given graph is NP-complete for all $k\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18880 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Majority additive coloring and the maximum degree Brause, Christoph Rautenbach, Dieter Schwartze, Laurin Combinatorics Kamyczura introduced the notion of a majority additive $k$-coloring of a graph $G$ as a function $c: V(G) \to \{1,2,\ldots,k\}$ such that $$\left|\left\{u \in N_G(v):\sum_{w \in N_G(u)} c(w) = s \right\}\right|\leq \max\left\{1,\frac{d_G(v)}{2}\right\}$$ for every vertex $v$ of $G$ and every positive integer $s$. We show that every graph $G$ of maximum degree $Δ$ admitting a majority additive coloring has a majority additive $\mathcal{O}\left(Δ^2\right)$-coloring. Under additional restrictions we improve this to sublinear in $Δ$. We show that determining whether a majority additive $k$-coloring exists for a given graph is NP-complete for all $k\geq 2$. |
| title | Majority additive coloring and the maximum degree |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.18880 |