Polychrome labelings of trees and cycles (Q2761009)

For \(G=(V,E)\) a finite connected simple graph and \(f\) a bijection between \(V\) and an abelian group \((A,+)\), the author calls \(f\) a polychrome labeling of \(G\) by \(A\) if all sums \(f(u)+f(v)\), where \(uv\) runs through the edges in \(E\), are distinct. This concept is related to graceful, harmonious, and elegant labelings of graphs. Results on polychrome labelings of paths and cycles are proved; we mention some of them. If \(A\) is a product of one or more odd cyclic groups and one even cyclic group, then the path \(P_{|A|}\) has a polychrome labeling by \(A\). \(P_{2^d}\), \(d>1,\) has no polychrome labeling by the product of \(d\) \(Z_2\). If \(B\) has odd order \(m\) and \(k>1\), then the cycle of \(2^{k+1}m\) vertices has a polychrome labeling by \(Z_{2}\times Z_{2^k}\times B\). It is conjectured that every tree on \(n\) vertices has a polychrome labeling by \(Z_n\).