Trees of odd order with at most two vertices of degree two are edge-graceful

Abstract

A graph G with q edges and p vertices is edge-graceful if some bijection f from E(G) onto {1,...,q} makes the induced vertex sums f^+(v), the sum of f(e) over the edges e incident to v, distinct modulo p. Lee conjectured in 1989 that every tree of odd order is edge-graceful; the broadest general result we have located, due in equivalent form to Kaplan, Lev and Roditty, covers trees of odd order with at most one vertex of degree two. We prove that every tree of odd order with at most two vertices of degree two is edge-graceful. The proof combines zero-sum block partitions of Z_n with perfect and hooked Langford sequences; the residual case analysis is verified symbolically for all odd n <= 5001 and holds uniformly beyond, and the construction was executed and independently re-checked on all 2,245,070 trees of odd order at most 25 with exactly two vertices of degree two. Since an edge-graceful graph is antimagic, the theorem also enlarges the family of trees known to be antimagic.