On Newman’s Conjecture and Prime Trees

In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman’s coprime mapping conjecture: If n is a positive integer and I is a set of n consecutive integers, then there is a bijection f :{1, 2, . . . , n}→ I such that gcd(i, f(i)) = 1 for 1 ≤ i ≤ n. The function f described in their theorem is called a coprime mapping. Around the same time, Roger Entringer conjectured that all trees are prime; that is, that if T is a tree with vertex set V , then there is a bijection L : V → {1, 2, . . . , |V |} such that gcd(L(x), L(y)) = 1 for all adjacent vertices x and y in V. So far, little progress towards a proof of this conjecture has been made. In this paper, we extend Pomerance and Selfridge’s theorem by replacing I with a set S of n integers in arithmetic progression and determining when there exist coprime mappings f : {1, 2, . . . , n} → S and g : {1, 3, . . . , 2n − 1} → S. We devote the rest of the paper to using coprime mappings to prove that various families of trees are prime, including palm trees, banana trees, binomial trees, and certain families of spider colonies.