Graphs: Graceful, Equitable and Distance Labelings

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. Juan Paul Aaron Marc Christina América 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 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. Model: vertices correspond to transmitters; use usual graph distance.

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

5. 1 4 7 2 Radio: 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. 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)

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)

9. w z v Counting Forbidden Values d(u,v)+ | f(u)-f(v) | ≥ 5

10. gap Using Gaps • 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).

11. gap + 2 2gap + 2 gap + 1 2gap + 1 1 2 gap gap gap Using Gaps, cont. • Have f(w) – f(u) ≥ ½(3 diam(G) + 3 – M) = gap. • If |V(G)| = n, this yields

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. G8 v1 w2 w1 v2 v8 x9 x5 x1 w3 w8 x10 x16 x2 x8 v7 v3 z x15 x11 x0 w4 w7 x6 x4 v6 v4 x14 x12 w6 w5 v5 x7 x3 x13 Using an intermediate labeling

14. 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.

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.

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. 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. 3 3 2 1 0 1 2 4 5 4 5 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.

19. 0 2 1 1 2 0 2 1 0 1 2 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}. k = 3: #V0 = #V1 = #V2 = 2 #E0+1 = #E1 = #E2 = 2

20. What’s known? graceful k-equitable • 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. • 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

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. Reading to get started • Radio labeling: Chartrand, Erwin, Harary, and Zhang, Radio labelings of graphs, Bull. Inst. Combin. Appl., 33 (2001), 77-85. • L(2,1) labeling: Griggs & Yeh, Labeling graphs with a condition at distance 2, SIAM J. Disc. Math., 5 (1992), 586-595. • Graceful & k-equitable labeling: Cahit, Equitable Tree Labellings, Ars. Combin. 40 (1995), 279-286. • General survey: Gallian, A dynamic survey of graph labeling, Dynamical Surveys, DS6, Electron. J. Combin. (1998). URL for these slides: http://faculty.csuci.edu/cynthia.wyels

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)

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.