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

2026-08-08

AI Summary

This study addresses the problem of determining whether, given a tree and two distinct real weights \(a\) and \(b\), there exists an edge labeling assigning each edge either \(a\) or \(b\) such that the sums of incident edge weights at every pair of adjacent vertices are distinct. By leveraging structural graph analysis, matching theory, and parity-based component decomposition, the authors provide the first complete characterization of tree structures that admit no such labeling across all possible weight configurations—including the cases \(a + b = 0\), \(ab = 0\), and the general setting. Building on this characterization, they devise a linear-time algorithm to decide the existence of a valid labeling and construct one when it exists. The results further establish that, apart from a few specific exceptional structures, almost all trees admit a valid assignment.

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.

Problem

• edge-weighting
• proper coloring
• trees
• graph labeling
• bipartition

Innovation

• edge-weighting
• tree
• structural characterization
• linear-time algorithm
• bipartition