素数倍数缺失图
Prime Multiple Missing Graphs
AI总结：
针对哥德巴赫猜想相关的图论研究，引入素数倍数缺失图，证明其连通性、直径上界及哈密顿性，并发现其与近哥德巴赫图的交集性质，为理解近哥德巴赫图结构提供基础。
摘要：
著名的哥德巴赫猜想已悬而未决近三个世纪。近期，哥德巴赫图被引入，以将该问题与图论文献关联起来。研究表明，这些图的连通性等价于猜想的肯定答案。一些经修改的图版本（如近哥德巴赫图）被证明在顶点数较少时是哈密顿图。在此背景下，我们引入一类名为素数倍数缺失图的图，使得近哥德巴赫图是这些图的有限交集。我们针对素数3、5及一般奇素数p研究这些图，证明它们是连通的且直径至多为3，同时对于偶数（>2）个顶点是哈密顿图。接着研究素数3和5对应的素数倍数缺失图的交集，证明它们是连通的且直径至多为4，对于偶数（>2）个顶点也为哈密顿图。我们观察到，有限哥德巴赫图和近哥德巴赫图的直径在顶点数不超过10000时被5所界。我们认为，通过大数据分析进一步研究这些图将有助于理解近哥德巴赫图的结构。
英文摘要：
The famous Goldbach conjecture remains open for nearly three centuries. Recently Goldbach graphs are introduced to relate the problem with the literature of Graph Theory. It is shown that the connectedness of the graphs is equivalent to the affirmative answer of the conjecture. Some modified version of the graphs, say, near Goldbach graphs are shown to be Hamiltonian for small number of vertices. In this context, we introduce a class of graphs, namely, prime multiple missing graphs such that near Goldbach graphs are finite intersections of these graphs. We study these graphs for primes 3,5 and in general for any odd prime p. We prove that these graphs are connected with diameter at most 3 and Hamiltonian for even (>2) vertices. Next the intersection of prime multiple missing graphs for primes 3 and 5 are studied. We prove that these graphs are connected with diameter at most 4 and they are also Hamiltonian for even (>2) vertices. We observe that the diameters of finite Goldbach graphs and near Goldbach graphs are bounded by 5 (up to 10000 vertices). We believe further study on these graphs with big data analysis will help to understand structures of near Goldbach graphs.
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