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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.17645 |
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| _version_ | 1866914408176812032 |
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| author | Liu, Feng Sun, Shuang Wang, Yan |
| author_facet | Liu, Feng Sun, Shuang Wang, Yan |
| contents | A graph is $\mathrm{ISK}_4$-free if it contains no induced subdivision of $K_4$. Lévêque et al. [\emph{J. Combin. Theory Ser. B} \textbf{102} (2012) 924--947] conjectured that all $\mathrm{ISK}_4$-free graphs are 4-colorable. Chen et al. [\emph{J. Graph Theory} \textbf{96} (2021) 554--577] proved that $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs are 4-colorable and asked whether such graphs are 3-colorable, where a diamond is $K_4$ minus one edge and a bowtie consists of two triangles sharing a vertex. In this paper, we characterize the structures of $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs and prove that such graphs are 3-colorable, which answers a question of Chen et al. [\emph{J. Graph Theory} \textbf{96} (2021) 554--577] affirmatively and extends a result of Chudnovsky et al. [\emph{J. Graph Theory} \textbf{92} (2019) 67--95]. Furthermore, our structural theorem yields a polynomial-time algorithm for decomposing $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs, and consequently a polynomial-time algorithm for coloring this class of graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_17645 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the structures of {diamond, bowtie}-free graphs that do not contain an induced subdivision of $K_4$ Liu, Feng Sun, Shuang Wang, Yan Combinatorics A graph is $\mathrm{ISK}_4$-free if it contains no induced subdivision of $K_4$. Lévêque et al. [\emph{J. Combin. Theory Ser. B} \textbf{102} (2012) 924--947] conjectured that all $\mathrm{ISK}_4$-free graphs are 4-colorable. Chen et al. [\emph{J. Graph Theory} \textbf{96} (2021) 554--577] proved that $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs are 4-colorable and asked whether such graphs are 3-colorable, where a diamond is $K_4$ minus one edge and a bowtie consists of two triangles sharing a vertex. In this paper, we characterize the structures of $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs and prove that such graphs are 3-colorable, which answers a question of Chen et al. [\emph{J. Graph Theory} \textbf{96} (2021) 554--577] affirmatively and extends a result of Chudnovsky et al. [\emph{J. Graph Theory} \textbf{92} (2019) 67--95]. Furthermore, our structural theorem yields a polynomial-time algorithm for decomposing $\{\mathrm{ISK}_4, \mathrm{diamond}, \mathrm{bowtie}\}$-free graphs, and consequently a polynomial-time algorithm for coloring this class of graphs. |
| title | On the structures of {diamond, bowtie}-free graphs that do not contain an induced subdivision of $K_4$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.17645 |