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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. Juan. Paul. Aaron. Marc. Christina.
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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
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.
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.
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 ______________
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.
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)
Undergraduate Contributions • Complete graphs, complete bipartite graphs, wheels • Gear graphs • Generalized prism graphs • Products of cycles
Strategies for establishing a lower bound for rn(G) • Counting “forbidden values” (e.g. bipartite graphs, wheels, gears) • Using “gaps” (vertex-transitive graphs)
w z v Counting Forbidden Values d(u,v)+ | f(u)-f(v) | ≥ 5
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).
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
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.
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
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.
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.
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
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.
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.
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
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
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.
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
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)
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.
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.