On the characterization of some algebraically defined bipartite graphs of girth eight
On the characterization of some algebraically defined bipartite graphs of girth eight
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关于周长为 8 的一些代数定义的二部图的表征
DOI：
10.1016/j.dam.2021.09.006
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Yuansheng Tang
中科院分区：
文献类型：
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作者：
Mingyao Xu;Xiaoyan Cheng;Yuansheng Tang
对任意域F和多项式f2，f3 ∈ F [x，y]，令（f2，f3）表示顶点划分为P <$L的二部图，其中P和L是F3的两个副本，（p1，p2，p3）∈ P与[l1，l2，l3]∈ L相邻当且仅当p2 + l2 = f2（p1，l1）且p3 + l3 = f3（p1，l1）.已知图Γ 3（F）= Γ F（xy，xy 2）围长为8.当F= Fq是奇特征有限域或F= F∞是特征为零的代数闭域时，证明了图Γ 3（F）是直到同构的所有图中围长至少为8的唯一图.这一猜想在f2，f3都是Fq上的单项式的情况下以及在f2，f3中至少有一个是F∞上的单项式的情况下得到了证实.如果f2，f3 ∈ Fq [x，y]中有一个是单项式，则证明了存在一个正整数M使得G= Γ FqM（f2，f3）同构于Γ 3（FqM），只要G的围长至少为8.本文证明了当多项式f2，f3的限制进一步放宽到其中一个是两个一元多项式的乘积时，这些结果是有效的。此外，所有这样的多项式f2，f3的特征完全。
For any field F and polynomials f 2, f 3∈ F [x, y], let Γ F (f 2, f 3) denote the bipartite graph with vertex partition P∪ L, where P and L are two copies of F 3, and (p 1, p 2, p 3)∈ P is adjacent to [l 1, l 2, l 3]∈ 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 (F)= Γ F (x y, x y 2) is known to be of girth eight. When F= F q is a finite field of odd characteristic or F= F∞ is an algebraically closed field of characteristic zero, the graph Γ 3 (F) is conjectured to be the unique one with girth at least eight among those Γ 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 F q, and for the case that at least one of f 2, f 3 is a monomial over F∞. If one of f 2, f 3∈ F q [x, y] is a monomial, it has also been proved the existence of a positive integer M such that G= Γ F q M (f 2, f 3) is isomorphic to Γ 3 (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.