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.

翻译：著名的哥德巴赫猜想已悬而未决近三个世纪。近期引入的哥德巴赫图将该问题与图论文献联系起来，已证明该图的连通性等价于猜想的肯定性解答。该图的一些修正版本（例如近哥德巴赫图）被证明在顶点数较少时具有哈密顿性。在此背景下，我们引入一类新图——素数倍数缺失图，使得近哥德巴赫图成为这类图的有限交。我们针对素数3、5及一般奇素数p研究此类图，证明这些图具有连通性（直径不超过3）且在偶数顶点（>2）时具有哈密顿性。继而研究素数3与5对应的素数倍数缺失图的交集，证明这些图具有连通性（直径不超过4）且在偶数顶点（>2）时同样具有哈密顿性。我们观察到有限哥德巴赫图与近哥德巴赫图的直径上限为5（在10000个顶点范围内）。我们相信通过大数据分析对这些图进行深入研究，将有助于理解近哥德巴赫图的结构特性。