تقرير
On the girth cycles of the bipartite graph $D(k,q)$
| العنوان: | On the girth cycles of the bipartite graph $D(k,q)$ |
|---|---|
| المؤلفون: | Xu, Ming, Cheng, Xiaoyan, Tang, Yuansheng |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics |
| الوصف: | For integer $k\geq2$ and prime power $q$, the algebraic bipartite graph $D(k,q)$ proposed by Lazebnik and Ustimenko (1995) is meaningful not only in extremal graph theory but also in coding theory and cryptography. This graph is $q$-regular, edge-transitive and of girth at least $k+4$. For its exact girth $g=g(D(k,q))$, F\"{u}redi et al. (1995) conjectured $g=k+5$ for odd $k$ and $q\geq4$. This conjecture was shown to be valid in 2016 when $(k+5)/2$ is the product of an arbitrary factor of $q-1$ and an arbitrary power of the characteristic of $\mathbb{F}_q$. In this paper, we determine all the girth cycles of $D(k,q)$ for $3\leq k\leq 5$, $q>3$, and those for $3\leq k\leq8$, $q=3$. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2207.12752 |
| رقم الأكسشن: | edsarx.2207.12752 |
| قاعدة البيانات: | arXiv |
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