Prime power and prime product distance graphs

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 kth-power graph (resp., strict prime kth-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 jth power of a prime for j <= k (resp., j = k). We prove that ppn(K-n) = inverted right perpendicular log(2)(n) inverted left perpendicular - 1, and for a nonempty k-chromatic graph G, ppn(G) = inverted right perpendicular log(2)(k) inverted left perpendicular - 1 or ppn(G) = inverted right perpendicular log(2)(k) inverted left perpendicular. We determine ppn(G) for all complete bi-, 3-, and 4-partite graphs. We prove that K-n is a prime kth-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 kth-power graph. In Theorems 2.4, 2.6, and 3.3, we relate prime product and prime power distance graphs to the Green-Tao Theorem, the Twin Prime Conjecture, and Fermat's Last Theorem. (C) 2018 Elsevier B.V. All rights reserved.