Computational techniques have been presented that will identify a graceful labeling for a given graph provided one exists, thereby confirming that the graph is indeed graceful. Supported by the necessary theory to guarantee a solution, these routines primarily rely upon constrained iterative methods and are often quite computationally expensive. A number of other methods for graceful labelings have been proposed, including those employing deterministic backtracking and tabu search, among others. Here, a genetic algorithm-inspired, metaheuristic search technique to attempt to ascertain graceful labels, via a modified objective functional, that operates on simple graphs is presented. This broad-spectrum method will be discussed and compared to previous techniques.

1. J. Ernstberger, A.D. Perkins
16 March 2013
Southeast MAA
A Metaheuristic Search
Technique for Graceful
Labels of Graphs
jernstberger@lagrange.edu
perkins@cse.msstate.edu

2. ● Rosa [7] denes 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
the resulting edge labelings are distinct."
Graceful Labels and Graphs

3. Graceful Labels and Graphs, cont.
● Applications of graph labelings (including graceful
labelings) are given in Bloom and Golomb [1].
● A graph that can be characterized via a graceful
label is said to be a graceful graph

4. Graceful Label, Examples
Figure 1: (Left) A tree diagram with five edges. (Right) A
wheel diagram with eight edges.

5. Another Tree Example, 35 Edges

6. Past Work
Ringel-Kotzig Conjecture - "all trees are graceful"
● 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.

7. Metaheuristic Approach
Holland[4] defines this concept of a genetic algorithm.
● A population P of m trial solutions (each with n
characteristics) is randomly created.
● 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.

8. Metaheuristic Approach, cont.
Image courtesy of National Geographic.
● Offspring are created via the processes mutation
and crossover.
o Mutation is the result of random noise being added to a
population (or individual attributes, the genes).
o Crossover occurs with a probability p and is a direct
swap between genes.
Comparison Example: Zebras

9. Population Evolution
Courtesy of https://www.indexdata.com/sites/indexdata.com/files/images/zebra.png

10. Population Evolution, cont.
12 5
94
17
83
11102 6

11. Population Evolution, (3) - SotF
12 5
94
17
83
11102 6

12. Population Evolution, (4) - SotF
Fast
Disease
Resistant
Better
Stripes
Fast
Endurance
Disease
Resistant

13. Population Evolution, (5) - Next Gen!
Fast
Disease
Resistant
Better
Stripes
Fast
Endurance
Disease
Resistant
F,E DR,F
FF DR,F
S,F S,DR

14. ● 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 has m members, on (0,q).
● Corresponding to the population was ,
where E has m members, each on (1,q).
● Each row of E is the computed labeling for the
edges in accordance to the related edge list.
● Practiced "elitism" with varying numbers or
percentages of the elite.
● In our formulation, mutation over the integers and
crossover were equivalent to a swap.
Metaheuristic Approach

15. Metaheuristic Approach, cont.
● The ith member of the population was
evaluated according to a fitness functional
● Objective is maximize J on (0,1).

16. Metaheuristic Approach, (3)
● 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.
● Trials (for each graph): 100 trials on each of 100
different graphs.

17. Results
Table 1: Comparison of GA data to the Eshghi, et. al.
ACO[5] and mathematical programming[2] routines.

18. Results, Tree
Table 2: Metaheuristic search for graceful labelings of
trees of size greater than 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.

19. 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

20. References
● 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.

21. References, cont.
● J.H. Holland. Genetic algorithms and the optimal
allocation of trials. SIAM Journal of Computing,
2(2), 1973.
● 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.

22. Refences, (3)
● 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.