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Bounding the number of arithmetical structures on graphs
Discrete Mathematics ( IF 0.8 ) Pub Date : 2021-06-08 , DOI: 10.1016/j.disc.2021.112494
Christopher Keyes , Tomer Reiter

Let G be a connected undirected graph on n vertices with no loops but possibly multiedges. Given an arithmetical structure (r,d) on G, we describe a construction which associates to it a graph G on n1 vertices and an arithmetical structure (r,d) on G. By iterating this construction, we derive an upper bound for the number of arithmetical structures on G depending only on the number of vertices and edges of G. In the specific case of complete graphs, possibly with multiple edges, we refine and compare our upper bounds to those arising from counting unit fraction representations.



中文翻译:

限制图上算术结构的数量

Gn个顶点上的无环连通无向图,但可能有多重边。给定算术结构(r,d)G 上,我们描述了一个与图相关联的结构Gn-1 顶点和算术结构 (r,d)G. 通过迭代这种结构,我们推导的上界算术结构上的数字ģ仅在顶点和的边的数量取决于ģ。在完整图的特定情况下,可能具有多条边,我们将我们的上限与计数单位分数表示产生的上限进行细化和比较。

更新日期:2021-06-08
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