Strong majority colorings of graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911709293182976 |
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| author | Kalinowski, Rafał Kamyczura, Mateusz Pilśniak, Monika Woźniak, Mariusz |
| author_facet | Kalinowski, Rafał Kamyczura, Mateusz Pilśniak, Monika Woźniak, Mariusz |
| contents | Motivated by majority vertex-colorings of graphs and digraphs and majority edge-colorings of graphs, we introduce two concepts of strong majority colorings. A strong majority vertex-coloring of a graph $G=(V,E)$ is a mapping $c:V\rightarrow C$ such that for every vertex $v\in V$ and every color $α\in C$, at most half of the neighbors of $v$ have color $α$. The strong majority number of $G$, denoted Maj$(G)$, is the least number of colors in such a coloring. We show that Maj$(G)$ can be arbitrarily large and prove a tight upper bound Maj$(G)\le 2Δ(G)+1$ for every graph $G$ without pendant vertices. A strong majority edge-coloring of a graph $G$ is a mapping $c:E\rightarrow C$ such that for every edge $e\in E$ and every color $α\in C$, at most half of the edges adjacent to $e$ have color $α$. The strong majority index of $G$, denoted Maj'$(G)$, is the least number of colors in such a coloring. It is shown that there is an upper constant bound for Maj'$(G)$ of all admissible graphs $G$. We conjecture that this constant is as small as 4 and confirm this conjecture for numerous graph classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23828 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong majority colorings of graphs Kalinowski, Rafał Kamyczura, Mateusz Pilśniak, Monika Woźniak, Mariusz Combinatorics 05C15, 05C70 Motivated by majority vertex-colorings of graphs and digraphs and majority edge-colorings of graphs, we introduce two concepts of strong majority colorings. A strong majority vertex-coloring of a graph $G=(V,E)$ is a mapping $c:V\rightarrow C$ such that for every vertex $v\in V$ and every color $α\in C$, at most half of the neighbors of $v$ have color $α$. The strong majority number of $G$, denoted Maj$(G)$, is the least number of colors in such a coloring. We show that Maj$(G)$ can be arbitrarily large and prove a tight upper bound Maj$(G)\le 2Δ(G)+1$ for every graph $G$ without pendant vertices. A strong majority edge-coloring of a graph $G$ is a mapping $c:E\rightarrow C$ such that for every edge $e\in E$ and every color $α\in C$, at most half of the edges adjacent to $e$ have color $α$. The strong majority index of $G$, denoted Maj'$(G)$, is the least number of colors in such a coloring. It is shown that there is an upper constant bound for Maj'$(G)$ of all admissible graphs $G$. We conjecture that this constant is as small as 4 and confirm this conjecture for numerous graph classes. |
| title | Strong majority colorings of graphs |
| topic | Combinatorics 05C15, 05C70 |
| url | https://arxiv.org/abs/2605.23828 |