On Scott's odd induced subgraph conjecture and a related problem
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915947836604416 |
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| author | Ning, Bo |
| author_facet | Ning, Bo |
| contents | For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $χ(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2χ(G))$ for every graph without isolated vertices, and conjectured that the factor $2$ can be removed. Wang and Wu (JGT, 2024) showed that this conjecture fails for bipartite graphs, but holds for line graphs. In this article, we confirm Scott's conjecture for claw-free graphs without isolated vertices, thereby strengthening the result of Wang and Wu. We also construct $K_{1,r}$-free graphs of arbitrarily large order to show that the conjecture fails for this broader class, for every integer $r \ge 4$.
Wang and Wu also asked whether $f_o(L(G)) \ge n/2$ holds for every connected regular graph $G$ of order $n \ge 3$. We show that $C_5$ is the smallest counterexample to this problem. On the positive side, we prove that if $G$ is a connected $k$-regular $C_5$-free graph on $n$ vertices with $k \ge 2$, then $f_o(L(G)) \ge n/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19727 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Scott's odd induced subgraph conjecture and a related problem Ning, Bo Combinatorics For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $χ(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2χ(G))$ for every graph without isolated vertices, and conjectured that the factor $2$ can be removed. Wang and Wu (JGT, 2024) showed that this conjecture fails for bipartite graphs, but holds for line graphs. In this article, we confirm Scott's conjecture for claw-free graphs without isolated vertices, thereby strengthening the result of Wang and Wu. We also construct $K_{1,r}$-free graphs of arbitrarily large order to show that the conjecture fails for this broader class, for every integer $r \ge 4$. Wang and Wu also asked whether $f_o(L(G)) \ge n/2$ holds for every connected regular graph $G$ of order $n \ge 3$. We show that $C_5$ is the smallest counterexample to this problem. On the positive side, we prove that if $G$ is a connected $k$-regular $C_5$-free graph on $n$ vertices with $k \ge 2$, then $f_o(L(G)) \ge n/2$. |
| title | On Scott's odd induced subgraph conjecture and a related problem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.19727 |