Graphs: Graceful, Equitable and Distance Labelings

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Presentation transcript:

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

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.

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.

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)

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.

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.

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) Maybe verbal explanation of vertex-transitive

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”

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

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

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.

Using an intermediate labeling v3 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

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)

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.

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

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.

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

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

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.

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.

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), 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) Get dates, have history a little more tip-of-tongue

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.