A survey on Hedetniemi's conjecture
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912248088231936 |
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| author | Zhu, Xuding |
| author_facet | Zhu, Xuding |
| contents | In 1966, Hedetniemi conjectured that for any positive integer $n$ and graphs $G$ and $H$, if neither $G$ nor $H$ is $n$-colourable, then $G \times H$ is not $n$-colourable. This conjecture has received significant attention over the past half century, and was disproved by Shitov in 2019. Shitov's proof shows that Hedetniemi's conjecture fails for sufficiently large $n$. Shortly after Shitov's result, smaller counterexamples were found in a series of papers, and it is now known that Hedetniemi's conjecture fails for all $n \ge 4$, and holds for $n \le 3$. Hedetniemi's conjecture has inspired extensive research, and many related problems remain open. This paper surveys the results and problems associated with the conjecture, and explains the ideas used in finding counterexamples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16078 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A survey on Hedetniemi's conjecture Zhu, Xuding Combinatorics History and Overview In 1966, Hedetniemi conjectured that for any positive integer $n$ and graphs $G$ and $H$, if neither $G$ nor $H$ is $n$-colourable, then $G \times H$ is not $n$-colourable. This conjecture has received significant attention over the past half century, and was disproved by Shitov in 2019. Shitov's proof shows that Hedetniemi's conjecture fails for sufficiently large $n$. Shortly after Shitov's result, smaller counterexamples were found in a series of papers, and it is now known that Hedetniemi's conjecture fails for all $n \ge 4$, and holds for $n \le 3$. Hedetniemi's conjecture has inspired extensive research, and many related problems remain open. This paper surveys the results and problems associated with the conjecture, and explains the ideas used in finding counterexamples. |
| title | A survey on Hedetniemi's conjecture |
| topic | Combinatorics History and Overview |
| url | https://arxiv.org/abs/2502.16078 |