Proper $\{a,b\}$-edge-weightings of trees

Abstract

Let $a$ and $b$ be distinct real weights. An $\{a,b\}$-edge-weighting of a tree assigns one of these weights to each edge and is proper if adjacent vertices have different sums of incident edge weights. For every such pair, we give an explicit structural characterization of the trees that do not admit a proper $\{a,b\}$-edge-weighting. If $ab(a+b)\neq0$, then $K_2$ is the only tree without such a weighting. If $a+b=0$, then a tree has no proper $\{a,b\}$-edge-weighting exactly when every vertex has degree $1$ or $3$ and the subgraph induced by the degree-$3$ vertices has a perfect matching. For the remaining case $ab=0$, form the spanning forest consisting of the edges whose deletion leaves two odd-order components. A tree $T$ has no proper $\{a,b\}$-edge-weighting exactly when both bipartition classes have odd order and every component of this forest satisfies two conditions. First, every component satisfies the preceding degree-and-matching condition. Second, within each component, the degree of a vertex $v$ in the forest plus twice the number of incident edges $e$ outside the forest for which the component of $T-e$ not containing $v$ has an odd number of vertices from each bipartition class is independent of $v$. For every fixed pair of distinct real weights, the proofs yield a linear-time algorithm that decides whether a proper $\{a,b\}$-edge-weighting exists and constructs one when it does.