Every Complete Binary Tree is Prime

A graph with a vertex set V is said to have a prime labeling if its vertices can be labeled with distinct integers 1; 2; ; jV j such that for every edge fx; yg, the labels assigned to x and y are relatively prime. A tree is prime if it has at least one prime labeling. Around 1980, Entringer conjectured that every tree is prime. After three decades, this conjecture remains open. Nevertheless, a few special classes of trees, specifically paths, stars, caterpillars, spiders, and small trees, have been shown to be prime. Among different types of trees, binary trees are probably the most frequently used in computer science. Fu and Huang showed that every perfect binary tree of order 2d . . 1 is prime. Although Fu and Huang ambiguously called perfect binary trees as complete binary trees in their paper, it has been verified that they only proved that perfect binary trees are prime. In this paper, the author looked beyond perfect binary trees and devised a two-step method to prove that every complete binary tree is prime. First, for the case of 2k . . 1 vertices, a near prime labeling was constructed such that the co-prime requirement was satisfied for every edge, except possibly for the edges between right leaves and their parents. In order to successfully construct a prime labeling without co-prime violations, the original prime labeling problem was transformed into a complete (co-prime) matching problem between the right leaves and their parents. By applying Hall's Theorem, we proved that a complete (co-prime) matching exists for the right leaves and their parents, thus proving that a prime labeling exists for every complete binary tree with 2k . . 1 vertices. Second, for the case of 2k vertices, we applied Bertrand-Chebyshev Theorem and