On the Chromatic Number of Grassmann Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909625917374464 |
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| author | D'haeseleer, Jozefien Taranchuk, Vladislav |
| author_facet | D'haeseleer, Jozefien Taranchuk, Vladislav |
| contents | In this paper we study the chromatic number of the Grassmann graphs $J_q(n, m)$. We show that $\binom{n-m+1}{1}_q \leq χ(J_q(n, m)) \leq \binom{n}{1}_q$, which is analogous to the best-known bounds for the chromatic number of the Johnson graphs $J(n, m)$. When $m = 2$, determining $χ(J_q(n, 2))$ is equivalent to determining the smallest number of partial line parallelisms that one can partition the lines of PG$(n-1, q)$ into. We survey known results about line parallelisms and their implications for $χ(J_q(n, 2))$. Finally, we prove that when $q$ is any power of two, and $n$ is any even integer, then $χ(J_q(n, 2)) < 2\binom{n-1}{1}_q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_22055 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Chromatic Number of Grassmann Graphs D'haeseleer, Jozefien Taranchuk, Vladislav Combinatorics 05B25 (Primary) 05B40, 52C17 (Secondary) In this paper we study the chromatic number of the Grassmann graphs $J_q(n, m)$. We show that $\binom{n-m+1}{1}_q \leq χ(J_q(n, m)) \leq \binom{n}{1}_q$, which is analogous to the best-known bounds for the chromatic number of the Johnson graphs $J(n, m)$. When $m = 2$, determining $χ(J_q(n, 2))$ is equivalent to determining the smallest number of partial line parallelisms that one can partition the lines of PG$(n-1, q)$ into. We survey known results about line parallelisms and their implications for $χ(J_q(n, 2))$. Finally, we prove that when $q$ is any power of two, and $n$ is any even integer, then $χ(J_q(n, 2)) < 2\binom{n-1}{1}_q$. |
| title | On the Chromatic Number of Grassmann Graphs |
| topic | Combinatorics 05B25 (Primary) 05B40, 52C17 (Secondary) |
| url | https://arxiv.org/abs/2505.22055 |