A Note on Set Graceful Labeling of Graphs

TL;DR

This paper proves a conjecture that characterizes when a complete bipartite graph can be labeled with subsets of a set so that all symmetric differences of adjacent vertices cover all nonempty subsets, showing only stars qualify.

Contribution

It provides a complete proof of a conjecture linking set labelings and graph structure, specifically characterizing stars via symmetric difference conditions.

Findings

Only star graphs satisfy the set labeling condition.

The set of symmetric differences covers all nonempty subsets.

The conjecture is affirmatively settled.

Abstract

We settle affirmatively a conjecture posed in [S. M. Hegde, Set colorings of graphs, European Journal of Combinatorics 30 (4) (2009), 986--995]: If some subsets of a set X are assigned injectively to all vertices of a complete bipartite graph G such that the collection of all sets, each of which is the symmetric difference of the sets assigned to the ends of some edge, is the set of all nonempty subsets of X, then G is a star.