The document presents a metaheuristic search technique for graceful labelings of graphs, detailing definitions and previous methodologies including ant colony optimization and genetic algorithms. It describes a new approach utilizing random permutations of integers for population generation and evaluates fitness through a specific functional. Experimental results compare the efficiency of this method against traditional techniques, highlighting the challenges and future work in improving large graph labelings.

1. A Metaheuristic Search Technique for Graceful
Labels of Graphs
J. Ernstberger1 and A. D. Perkins2
5 March 2013
1
LaGrange College, jernstberger@lagrange.edu
2
Mississippi State University, perkins@cse.msstate.edu

2. Graceful Labels/Graphs
Rosa [7] defines the notion of a graceful label of a graph.
Restated by Vassilevska[8]
“A graceful labeling of a graph G with q edges is an
injection from the vertices of G to the set S of
integers {0, 1, . . . , q} such that when an edge with
vertices x and y are assigned the label |f (x) − f (y)|,
the resulting edge labelings are distinct.”
Applications of graph labelings (including graceful labelings)
are given in Bloom and Golomb [1].
Def. A graph that can be characterized via a graceful label is said
to be a graceful graph.

3. Graceful Label, Example
2
1
4 35
0 0
6
4
8
1
Figure: Graceful labelings of graphs. Left: A tree with six vertices.
Right: An eight-edged wheel graph.

4. Past Techniques
Eshghi and Azimi[2] - Constrained programming problem.
Fang[3] - simulated annealing (a statistical mechanics lens to
minimization) for graceful labelings of trees.
Eshghi and Mahmoudzadeh[5] - Metaheuristics (ant colony)
approach for graceful labelings.
Redl[6] - Integer and constrained programming problem with
specific implementation for speed.
Others

5. Metaheuristic Approach - Inspired by Genetic Algorithm
Holland[4] defines this concept of a genetic algorithm.
A population P of trial solutions is randomly created, typically
in Rm×n – m “solutions” for a problem whose domain is in
Rn(a, b).
A fitness function is defined so that the goodness-of-fit of
each member (possible solution) is measured.
Those solutions deemed most fit remain until a new
“generation”. This process is known as elitism.
Offspring are created via the two processes mutation and
crossover.
Mutation is the result of random noise being added to a
population (or individual attributes, the genes).
Crossover occurs with a probably p and is a direct swap
between genes.

6. Metaheuristic Approach, cont.
How did our formulation vary?
We use random permutations of the integers in the set
{0, 1, . . . , q} (q is the number of edges of the graph) to
create each member of the population. The population
P ∈ Zm×n(0, q)
Corresponding to the population was F(P) → E where
E ∈ Zm×q(1, q). Each row is the computed labeling for the
edges in accordance to the related edge list.
Practiced “elitism” with varying numbers/percentages of the
elite.
In our formulation, mutation over the integers and crossover
were equivalent–a swap.

7. Metaheuristic Approach, 3
The ith member of the population was evaluated according to
a fitness functional
J(Pi) =
1
q
q
j=1
(sort(F(Pi))j ≡ j)
=
1
q
q
j=1
(sort(Ei)j ≡ j)
Objective is to maximize, on (0,1), the fitness functional.
Notes:
There is no formal theory for the convergence (or lack
thereof) of the genetic algorithm.
The algorithm cannot state definitively that there is no
graceful label for a graph.

8. Experiment
Machinery
1. Intel Core i5 (2.6 GHz) / 4 GB RAM
2. Intel Core 2 Duo (3.0 GHz) / 2 GB RAM
3. All devices running MATLAB R2012B.
Trials - for each graph, T5, W10, etc., 100 trials on each of
100 different graphs.

9. Results
Graph Type Name Ant Col. Math Prog. GA
Wheels
W10 12.03 55.50 1.26
W15 139.37 3358.11 30.91
Helms
H8 22.40 1585.44 1.81
H10 37.71 3471.22 11.55
Cycles
C10 4.26 0.00 0.01
C15 166.26 0.65 0.28
Trees
T20 368.42 149.12 0.65
T25 1288.24 2898.14 2.82
Table: Comparison of GA data to the Eshghi, et. al. ACO[5] and
mathematical programming[2] routines.

10. Results, trees
Name Mean Gens. Prob. Conv. Mean Time(s)
T25 838.6 1.0000 2.82
T28 1603.3 0.9998 5.42
T30 2292.7 0.9991 9.04
T35 5823.0 0.9779 25.20
T40 11663.4 0.8267 58.56
Table: Metaheuristic search for graceful labelings of trees of size n ≥ 25.
Ratio of increase on time and mean generations more than
doubles (on 2.6x and 2.3x,resp.).
Due, in part, to the sort used currently.

11. Future Work
Explore labelings for large trees
Generalized Petersen graphs and product graphs
Computing-efficient fitness functional
Make software available
Port to computing-efficient language

12. References I
Gary S Bloom and Solomon W Golomb.
Applications of numbered undirected graphs.
Proceedings of the IEEE, 65(4):562–570, 1977.
Kourosh Eshghi and Parham Azimi.
Applications of mathematical programming in graceful labeling
of graphs.
Journal of Applied Mathematics, 2004(1):1–8, 2004.
Wenjie Fang.
A computational approach to the graceful tree conjecture.
arXiv preprint arXiv:1003.3045, 2010.
J.H. Holland.
Genetic algorithms and the optimal allocation of trials.
SIAM Journal of Computing, 2(2), 1973.

13. References II
Houra Mahmoudzadeh and Kourosh Eshghi.
A metaheuristic approach to the graceful labeling problem.
International Journal of Applied Metaheuristic Computing
(IJAMC), 1(4):42–56, 2010.
Timothy A Redl.
Graceful graphs and graceful labelings: two mathematical
programming formulations and some other new results.
Congressus Numerantium, pages 17–32, 2003.
Alexander Rosa.
On certain valuations of the vertices of a graph.
In Theory of Graphs (Internat. Symposium, Rome, pages
349–355, 1966.
Virginia Vassilevska.
Coding and graceful labeling of trees.