Prime Power and Prime Product Distance Graphs

TL;DR

This paper introduces prime product and prime power distance graphs, establishing their properties, calculating their parameters for various graph classes, and exploring connections to famous number theory conjectures and theorems.

Contribution

It defines new classes of graphs based on prime factorizations of vertex label differences and determines their parameters for key graph families, linking graph theory with number theory.

Findings

Calculated prime product number for complete graphs and chromatic graphs.

Characterized prime power graphs among complete, bipartite, and cycle graphs.

Connected prime power distance graphs to major number theory conjectures.

Abstract

A graph $G$ is a $k$-prime product distance graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the product of at most $k$ primes. A graph has prime product number $ppn(G)=k$ if it is a $k$-prime product graph but not a $(k-1)$-prime product graph. Similarly, $G$ is a prime $k$th-power graph (respectively, strict prime $k$th-power graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the $j$th power of a prime, for $j \leq k$ (respectively, the $k$th power of a prime exactly). We prove that $ppn(K_n) = \lceil \log_2(n)\rceil - 1$, and for a nonempty $k$-chromatic graph $G$, $ppn(G) = \lceil \log_2(k)\rceil - 1$ or $ppn(G) = \lceil \log_2(k)\rceil$. We determine $ppn(G)$ for all complete bipartite, 3-partite, and 4-partite graphs. We prove that $K_n$ is a prime $k$th-power graph if and only if $n < 7$, and we determine conditions on cycles and outerplanar graphs $G$ for which $G$ is a strict prime $k$th-power graph. We find connections between prime product and prime power distance graphs and the Twin Prime Conjecture, the Green-Tao Theorem, and Fermat's Last Theorem.