Maximum odd induced subgraph of a graph concerning its chromatic number
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909118600577024 |
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| author | Wang, Tao Wu, Baoyindureng |
| author_facet | Wang, Tao Wu, Baoyindureng |
| contents | Let $f_{o}(G)$ be the maximum order of an odd induced subgraph of $G$. In 1992, Scott proposed a conjecture that $f_{o}(G)\geq \frac {n} {2χ(G)}$ for a graph $G$ of order $n$ without isolated vertices, where $χ(G)$ is the chromatic number of $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang and Wargo in 1997, which states that $f_{o}(G)\geq 2\lfloor\frac {n} {4}\rfloor$ for a connected graph $G$ of order $n$. Scott's conjecture is open for a graph with chromatic number at least 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_10895 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Maximum odd induced subgraph of a graph concerning its chromatic number Wang, Tao Wu, Baoyindureng Combinatorics 05C07, 05C15 Let $f_{o}(G)$ be the maximum order of an odd induced subgraph of $G$. In 1992, Scott proposed a conjecture that $f_{o}(G)\geq \frac {n} {2χ(G)}$ for a graph $G$ of order $n$ without isolated vertices, where $χ(G)$ is the chromatic number of $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang and Wargo in 1997, which states that $f_{o}(G)\geq 2\lfloor\frac {n} {4}\rfloor$ for a connected graph $G$ of order $n$. Scott's conjecture is open for a graph with chromatic number at least 3. |
| title | Maximum odd induced subgraph of a graph concerning its chromatic number |
| topic | Combinatorics 05C07, 05C15 |
| url | https://arxiv.org/abs/2211.10895 |