1. Graphs: Graceful, Equitable and Distance Labelings
Cindy Wyels
California State University Channel Islands
Graph theory Ideas for Undergraduate Research
MAA Invited Paper Session at MathFest, 2006
Organizer: Aparna Higgins, University of Dayton

2. URL for slides provided at end.
Juan
Aaron
Paul
Marc
América
Christina
Overview
Labeling schemes
Distance labeling schemes
Graceful and k-equitable labeling
Advantages for undergraduate research
Low faculty and student start-up “costs”
Lots of accessible open problems
Can “get hands dirty” quickly
REU context – discuss students’ backgrounds
URL for slides provided at end.

3. Distance Labeling Schemes
Motivating Context: assignment of channels to FM radio stations
General Idea: transmitters that are geographically close must be assigned channels with large frequency differences; transmitters that are further apart may receive channels with relatively close frequencies.
(All graphs are simple, connected graphs.) true for all graph labeling in general?
“Any graph I refer to is simple and connected.”
Model: vertices correspond to transmitters; use usual graph distance.

4. Some distance labeling schemes
f : V(G) → N satisfies ______________
Ld(2,1):
Ld(3,2,1), L(h,k), L(λ1, …λk): analogous
antipodal
Radio:
Antipodal: (same)
k-labeling: (same)

5. Radio: 1 4 7 2
The radio number of a graph G, rn(G), is the smallest integer m such that G has a radio labeling f with max{f(v) | v in V(G)} = m.
Don’t say “span”. 4 1 6 3
rn(P4) = 6.

6. Good problem: find rn(G) for all graphs G belonging to some graph family
“… determining the radio number seems a difficult problem even for some basic families of graphs.”
(Liu and Zhu)
Complete k-partite graphs (Chartrand, Erwin, Harary, Zhang)
Paths and cycles (Liu, Zhu)
Squares of paths and cycles (Liu, Xie)
Spiders (Liu, submitted)
Liu also has lower bounds for radio numbers of spiders – maybe done w/ spiders overall? Also working on caterpillars.

7. Undergraduate Contributions
Complete graphs, complete bipartite graphs, wheels
Gear graphs
Generalized prism graphs
Products of cycles

8. Strategies for establishing a lower bound for rn(G)
Counting “forbidden values” (e.g. bipartite graphs, wheels, gears)
Using “gaps” (vertex-transitive graphs)
Maybe verbal explanation of vertex-transitive

9. Counting Forbidden Values
d(u,v)+ | f(u)-f(v) | ≥ 5
Vertex type
Minimum label diff
Min. # of forbidden values
# of values used as labels z 3 2 1 w n v
2(n -1)
n -1
Total:
2n +1 z w v
Say something about what the gear graph is before diving in.
Say something about smallest and largest labels… help w/ “min. # of forbidden values”

10. Using Gaps gap Need lemma giving M = max{d(u,v)+d(v,w)+d(w,v)}.
Assume f(u) < f(v) < f(w).
Summing the radio condition
d(u,v) + |f(u) - f(v)| ≥ diam(G) + 1
for each pair of vertices gives
M + 2f(w) – 2f(u) ≥ 3 diam(G) + 3
i.e.
f(w) – f(u) ≥ ½(3 diam(G) + 3 – M).
Last formula?
Label 2nd eq’n “gap”
Include number line illustration
gap

11. Using Gaps, cont. Have f(w) – f(u) ≥ ½(3 diam(G) + 3 – M) = gap.
If |V(G)| = n, this yields
gap + 2
gap
2gap + 2
2gap + 1
gap
gap + 1 1 2
gap
Last formula?
Label 2nd eq’n “gap”
Include number line illustration

12. Strategies for establishing an upper bound for rn(G)
Define a labeling, prove it’s a radio labeling, determine the maximum label.
Might use an intermediate labeling that orders the vertices {x1, x2, … xs} so that f(xi) > f(xj) iff i > j.
Using patterns, iteration, symmetry, etc. to define a labeling makes it easier to prove it’s a radio labeling.

13. Using an intermediate labelingv3 G8 z v1 v2 v4 v5 v6 v7 v8 w1 w3 w2 w4 w7 w5 w8 w6
x11 x0 x9
x10
x12
x13
x14
x15
x16 x1 x2 x5 x6 x4 x3 x8 x7

14. Using patterns, iteration, etc
Using patterns, iteration, etc. to prove the labeling is a radio labeling
For products of cycles and for generalized prism graphs, the gap was about half the diameter.
This gives | f(xi) – f(xj) | ≥ diam(G) + 1 whenever j – i ≥ 4.
Also, | f(xi) – f(xj) | = | f(xi+k) – f(xj+k) |, so it suffices to show the radio condition holds for {x1, x2, x3, x4}.
Using symmetry in labeling also advantageous.
KEEP EMPHASIZING “WE”! (students)

15. Some Radio Labeling Questions
Find relationships between rn(G) and specific graph properties (e.g. connectivity, diameter, etc.).
Investigate radio numbers of various product graphs, and/or determine the relationship between the radio number of a product graph and the radio numbers of its factor graphs.
Investigate radio numbers of powers of graphs.
Determine properties of minimal labelings. E.g. is the radio number always realized by a labeling that assigns 1 to a cut vertex? … to a vertex of highest degree?
Create an algorithm for checking labelings.
Realign/ group the questions.

16. Find radio numbers of families of graphs
Generalized gears (adapt methods)
Ladders
Web graphs (products of cycles and paths)
Products of cycles of different sizes
Grid graphs (products of paths)
More generalized prisms

17. L(3,2,1) labeling Clipperton, Gehrtz, Szaniszlo,
undergraduates
L(3,2,1) labeling
Clipperton, Gehrtz, Szaniszlo,
and Torkornoo (2006) provide the
L(3,2,1)-labeling numbers for:
- Complete graphs - Complete bipartite graphs
- Paths - Cycles
- Caterpillars - n-ary trees
They also give an upper bound for the L(3,2,1)-labeling number in terms of the maximum degree of the graph.

18. Graceful and k-equitable labelings
Define a labeling f : V(G) → {0, 1, … |E(G)|}.
Edge (u,v) receives the label induced by | f(u) – f(v) |.
The labeling is graceful when none of the vertex or edge labels repeat. 3 4 1 2 5 2 1 3 5 4

19. Graceful and k-equitable labelings
Define a labeling f : V(G) → {0, 1, … k-1}.
Edge (u,v) receives the label induced by | f(u) – f(v) |.
Let #Vj and #Ej be the number of vertices and edges, respectively, labeled j.
The labeling is k-equitable if |#Vi - #Vj| ≤ 1 and
|#Ei - #Ej| ≤ 1 for all i ≠ j in {0, 1, … k-1}. 2 1
k = 3: 2 1
#V0 = #V1 = #V2 = 2
#E0+1 = #E1 = #E2 = 2

20. What’s known? graceful k-equitable Stars Paths Caterpillars
The Petersen graph
n-cycles for n ≡ 0, 3 (4)
Symmetric trees
All trees with no more than four leaves
All trees with no more than 27 vertices
Stars
Paths
Caterpillars
Eulerian graphs (conditions)
Cycles (conditions)
Wheels (k = 3)
All trees are 3-equitable.
All trees with fewer than five leaves are k-equit.
Faculty-student collab’s on 3-equitable trees
Jump from here to questions.

21. Some Graceful/ k-Equitable Questions
Investigate particular types of trees to determine whether they are k-equitable. (E.g. complete binary trees are currently under investigation.)
Explore whether particular families of graphs are k-equitable or graceful.
Investigate whether methods of “gluing” graceful trees together to form larger graceful trees extend to k-equitability.

22. URL for these slides: http://faculty.csuci.edu/cynthia.wyels
Reading to get started
Radio labeling: Chartrand, Erwin, Harary, and Zhang, Radio labelings of graphs, Bull. Inst. Combin. Appl., 33 (2001),
L(2,1) labeling: Griggs & Yeh, Labeling graphs with a condition at distance 2, SIAM J. Disc. Math., 5 (1992),
Graceful & k-equitable labeling: Cahit, Equitable Tree Labellings, Ars. Combin. 40 (1995),
General survey: Gallian, A dynamic survey of graph labeling, Dynamical Surveys, DS6, Electron. J. Combin. (1998).
URL for these slides:

24. Conjectures
Graceful labelings were defined in the context of graph decompositions.
Rosa’s Th’m: If a tree T with m edges has a graceful labeling, then K2m+1 decomposes into 2m+1 copies of T. (1968)
Kotzig-Ringel Conj: Every tree has a graceful labeling.
Cahit’s Conj: All trees are k-equitable. (1990)
Get dates, have history a little more tip-of-tongue

25. Are all complete-binary trees k-equitable?
Know true for k = 2^n, n = 0, 1, …, 5. Think method extends to all n.
Know true for k = 2, 3, 4, 5, 6, 7.
Need last step to show true for all k congruent to 0, 1 mod 4.
Why worry about complete binary trees? Would like to generalize any findings and methods.

26. All trees are 3-equitable
Outline how this was done. Emphasize student contributions!
Rosa’s Th’m: If a tree T with m edges has a graceful labeling, then K2m+1 decomposes into 2m+1 copies of T.
Kotzig-Ringel Conj: Every tree has a graceful labeling.
Cahit’s Conj: All trees are k-equitable.