Prime Multiple Missing Graphs

TL;DR

This paper introduces prime multiple missing graphs, a new class related to Goldbach graphs, demonstrating their connectivity and Hamiltonian properties, which may shed light on the Goldbach conjecture through graph theoretical analysis.

Contribution

It defines prime multiple missing graphs and explores their properties, including connectivity and Hamiltonicity, for various primes, linking them to Goldbach graphs and conjecture.

Findings

Prime multiple missing graphs are connected with diameter at most 3 for odd primes.

These graphs are Hamiltonian for even number of vertices greater than 2.

The diameters of Goldbach and near Goldbach graphs are bounded by 5 up to 10,000 vertices.

Abstract

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.