An Adiabatic Quantum Algorithm for Determining Gracefulness of A Graph

TL;DR

This paper introduces an adiabatic quantum algorithm to determine whether a graph is graceful and to find a graceful labelling, demonstrating polynomial time complexity for graphs with up to 15 vertices.

Contribution

The paper presents the first adiabatic quantum algorithm for testing graph gracefulness and finding graceful labellings, addressing a problem with limited classical solutions.

Findings

Algorithm successfully determines gracefulness of graphs up to 15 vertices.

Numerical simulations show polynomial time complexity.

Provides a quantum approach to a classical combinatorial problem.

Abstract

Graph labelling is one of the noticed contexts in combinatorics and graph theory. Graceful labelling for a graph $G$ with $e$ edges, is to label the vertices of $G$ with $0, 1, \cdots, e$ such that, if we specify to each edge the difference value between its two ends, then any of $1, 2, \cdots, e$ appears exactly once as an edge label. For a given graph, there is still few efficient classical algorithms that determines either it is graceful or not, even for trees - as a well-known class of graphs. In this paper, we introduce an adiabatic quantum algorithm, which for a graceful graph $G$ finds a graceful labelling. Also, this algorithm can determine if $G$ is not graceful. Numerical simulations of the algorithm reveal that its time complexity has a polynomial behaviour with the problem size up to the range of 15 qubits.