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arXiv 1912.04592math.CO
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围长为8的代数定义的二部图的特征

On the characterization of some algebraically defined bipartite graphs of girth eight

Ming Xu, Xiaoyan Cheng, Yuansheng Tang

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摘要：

对于任何域$\mathbb{F}$和多项式$f_{2},f_{3}\in\mathbb{F}[x,y]$，令$Γ_{\mathbb{F}}(f_{2},f_{3})$表示顶点划分为$P\cup L$的二部图，其中$P$和$L$是$\mathbb{F}^{3}$的两个副本，并且$(p_{1},p_{2},p_{3})\in P$与$[l_{1},l_{2},l_{3}]\in L$相邻当且仅当$p_{2}+l_{2}=f_{2}(p_{1},l_{1})$和$p_{3}+l_{3}=f_{3}(p_{1},l_{1})$。已知图$Γ_{3}(\mathbb{F})=Γ_{\mathbb{F}}(xy,xy^{2})$的围长为8。当$\mathbb{F}=\mathbb{F}_q$是奇数大小的有限域$q$或$\mathbb{F}=\mathbb{F}_{\infty}$是特征为零的代数闭域时，证明了图$Γ_{3}(\mathbb{F})$是直到同构的$Γ_{\mathbb{F}}(f_{2},f_{3})$中围长至少为8的唯一图.这一猜想在两个$f_{2},f_{3}$都是$\mathbb{F}_q$上的单项式的情况下，以及在至少一个$f_{2},f_{3}$是$\mathbb{F}_{\infty}$上的单项式的情况下得到了证实。如果$f_{2},f_{3}\in\mathbb{F}_q[x,y]$中的一个是单项式，则还证明了存在正整数$M$使得$G=Γ_{\mathbb{F}_{q^{M}}}(f_2,f_3)$同构于$Γ_{3}(\mathbb{F}_{q^{M}})$，只要$G$的围长至少为8。在本文中，这些结果是有效的多项式$f_2,f_3$的限制进一步放宽到其中之一是两个一元多项式的产品。此外，所有这样的多项式$f_2,f_3$的特征完全。

英文摘要：

For any field $\mathbb{F}$ and polynomials $f_{2},f_{3}\in\mathbb{F}[x,y]$, let $Γ_{\mathbb{F}}(f_{2},f_{3})$ denote the bipartite graph with vertex partition $P\cup L$, where $P$ and $L$ are two copies of $\mathbb{F}^{3}$, and $(p_{1},p_{2},p_{3})\in P$ is adjacent to $[l_{1},l_{2},l_{3}]\in L$ if and only if $p_{2}+l_{2}=f_{2}(p_{1},l_{1})$ and $p_{3}+l_{3}=f_{3}(p_{1},l_{1})$. The graph $Γ_{3}(\mathbb{F})=Γ_{\mathbb{F}}(xy,xy^{2})$ is known to be of girth eight. When $\mathbb{F}=\mathbb{F}_q$ is a finite field of odd size $q$ or $\mathbb{F}=\mathbb{F}_{\infty}$ is an algebraically closed field of characteristic zero, the graph $Γ_{3}(\mathbb{F})$ is conjectured to be the unique one with girth at least eight among those $Γ_{\mathbb{F}}(f_{2},f_{3})$ up to isomorphism. This conjecture has been confirmed for the case that both $f_{2},f_{3}$ are monomials over $\mathbb{F}_q$, and for the case that at least one of $f_{2},f_{3}$ is a monomial over $\mathbb{F}_{\infty}$. If one of $f_{2},f_{3}\in\mathbb{F}_q[x,y]$ is a monomial, it has also been proved the existence of a positive integer $M$ such that $G=Γ_{\mathbb{F}_{q^{M}}}(f_2,f_3)$ is isomorphic to $Γ_{3}(\mathbb{F}_{q^{M}})$ provided $G$ has girth at least eight. In this paper, these results are shown to be valid when the restriction on the polynomials $f_2,f_3$ is relaxed further to that one of them is the product of two univariate polynomials. Furthermore, all of such polynomials $f_2,f_3$ are characterized completely.

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