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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2510.00712 |
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| _version_ | 1866910036297515008 |
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| author | Mphako-Banda, Eunice Kriel, Christo Alochukwu, Alex |
| author_facet | Mphako-Banda, Eunice Kriel, Christo Alochukwu, Alex |
| contents | In this paper, we introduce and study a novel graph parameter called the $k$-defect number, denoted $ϕ_{k}(G)$, for a graph $G$ and an integer $0\leq k\leq |E(G)|$. Unlike traditional defective colorings that bound the local degree within monochromatic components, the $k$-defect number represents the smallest number of colors required to achieve a vertex coloring of $G$ having exactly \emph{$k$ monochromatic edges (also termed ``bad edges")}. This parameter generalizes the well-known chromatic number of a graph, $χ(G)$, which is precisely $ϕ_{0}(G)$. We establish fundamental properties of the $k$-defect number and derive bounds on $ϕ_{k}(G)$ for specific graph classes, including trees, cycles, and wheels. Furthermore, we extend and generalize several classical properties of the chromatic number to this new edge-centric $k$-defect framework for values of $1\leq k\leq |E(G)|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the $k$-defect number: Vertex Coloring with a Fixed Number of Monochromatic Edges Mphako-Banda, Eunice Kriel, Christo Alochukwu, Alex Combinatorics 05C15, 05C70 In this paper, we introduce and study a novel graph parameter called the $k$-defect number, denoted $ϕ_{k}(G)$, for a graph $G$ and an integer $0\leq k\leq |E(G)|$. Unlike traditional defective colorings that bound the local degree within monochromatic components, the $k$-defect number represents the smallest number of colors required to achieve a vertex coloring of $G$ having exactly \emph{$k$ monochromatic edges (also termed ``bad edges")}. This parameter generalizes the well-known chromatic number of a graph, $χ(G)$, which is precisely $ϕ_{0}(G)$. We establish fundamental properties of the $k$-defect number and derive bounds on $ϕ_{k}(G)$ for specific graph classes, including trees, cycles, and wheels. Furthermore, we extend and generalize several classical properties of the chromatic number to this new edge-centric $k$-defect framework for values of $1\leq k\leq |E(G)|$. |
| title | A note on the $k$-defect number: Vertex Coloring with a Fixed Number of Monochromatic Edges |
| topic | Combinatorics 05C15, 05C70 |
| url | https://arxiv.org/abs/2510.00712 |