Graphs: Graceful, Graphs: Graceful, Equitable and Distance Equitable and Distance

LabelingsLabelings

Cindy WyelsCindy Wyels

California State University Channel IslandsCalifornia State University Channel Islands

Graph theory Ideas for Undergraduate Research

MAA Invited Paper Session at MathFest, 2006

Organizer: Aparna Higgins, University of Dayton

OverviewOverview Labeling schemesLabeling schemes

• Distance labeling Distance labeling schemesschemes

• Graceful and Graceful and kk--equitable labelingequitable labeling

URL for slides provided at end.

Advantages for undergraduate researchAdvantages for undergraduate research• Low faculty and student start-up “costs”Low faculty and student start-up “costs”• Lots of accessible open problemsLots of accessible open problems• Can “get hands dirty” quicklyCan “get hands dirty” quickly

JuanAaron

Paul

Marc

América

Christina

Distance Labeling SchemesDistance 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.

Some distance labeling Some distance labeling schemesschemes

f : V(G) → N satisfies ______________

Ld(2,1):

Ld(3,2,1), L(h,k), L(λ1, …λk): analogous

2),(when

1),(when2)()(

vudd

vuddvfuf

Radio:

Antipodal: (same)

k-labeling: (same)

)(,1)(diam)()(),( GVvuGvfufvud

)(diam)()(),( Gvfufvud

1)()(),( kvfufvud

Radio: Radio: 1)(diam)()(),( Gvfufvud

4 1 6 3

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.

rn(P4) = 6.

Good problem: find Good problem: find rnrn((GG) for all ) for all graphs graphs GG belonging to some graph belonging to some graph

familyfamily

Complete Complete kk-partite graphs (Chartrand, Erwin, Harary, -partite graphs (Chartrand, Erwin, Harary, Zhang)Zhang)

Paths and cycles (Liu, Zhu)Paths and cycles (Liu, Zhu)

Squares of paths and cycles (Liu, Xie)Squares of paths and cycles (Liu, Xie)

Spiders (Liu, submitted)Spiders (Liu, submitted)

“… “… determining the radio number seems a difficult determining the radio number seems a difficult problem even for some basic families of graphs.” problem even for some basic families of graphs.”

(Liu and Zhu)(Liu and Zhu)

UndergraduateUndergraduate ContributionsContributions

Complete graphs, complete bipartite Complete graphs, complete bipartite graphs, wheelsgraphs, wheels

Gear graphsGear graphs

Generalized prism graphsGeneralized prism graphs

Products of cycles Products of cycles

Strategies for establishing a lower Strategies for establishing a lower bound for bound for rnrn((GG))

Counting “forbidden values” (e.g. Counting “forbidden values” (e.g. bipartite graphs, wheels, gears)bipartite graphs, wheels, gears)

Using “gaps” (vertex-transitive Using “gaps” (vertex-transitive graphs)graphs)

Counting Forbidden ValuesCounting Forbidden Values

Vertex Vertex typetype

MinimuMinimum label m label diffdiff

Min. # of Min. # of forbidden forbidden

valuesvalues

# of # of values values used as used as labelslabels

zz 33 22 11

ww 11 00 nn

vv 22 11 11

vv 22 2(2(n n --1)1)

n n -1-1

TotalTotal::

22n n +1+1 22n n +1+1

.4for24)( nnGrn n

d(u,v)+ | f(u)-f(v) | ≥ 5

z

w

v

7G

Need lemma giving Need lemma giving MM = max{ = max{dd((uu,,vv)+)+dd((vv,,ww)+)+dd((ww,,vv)}.)}.

Assume Assume ff((uu) < ) < ff((vv) < ) < ff((ww).).

Summing the radio condition Summing the radio condition

dd((uu,,vv) + |) + |ff((uu) - ) - ff((vv)| )| ≥ diam(≥ diam(GG) + 1) + 1

for each pair of vertices gives for each pair of vertices gives

MM + 2 + 2ff((ww) – 2) – 2ff((uu) ) ≥ 3 diam(≥ 3 diam(GG) + 3) + 3

i.e.i.e.

ff((ww) – ) – ff((uu) ) ≥ ½(3 diam(≥ ½(3 diam(GG) + 3 – ) + 3 – MM).).

Using GapsUsing Gaps

Have Have ff((ww) – ) – ff((uu) ) ≥ ½(3 diam(≥ ½(3 diam(GG) + 3 – ) + 3 – MM) = ) = gap.gap.

If |If |VV((GG)| = )| = nn, this yields, this yields

Using Gaps, cont.Using Gaps, cont.

gap + 1

gap + 2

gap

2gap + 22gap +

1

gap

1 2

gap

even.212

odd,12

1

)(n

ngap

nn

gapGrn

Strategies for establishing an upper Strategies for establishing an upper bound for bound for rnrn((GG))

Define a labeling, prove it’s a radio labeling, Define a labeling, prove it’s a radio labeling, determine the maximum label.determine the maximum label.

Might use an intermediate labeling that orders Might use an intermediate labeling that orders the vertices {the vertices {xx11, , xx22, … , … xxss} so that } so that ff((xxii) > ) > ff((xxjj) iff ) iff ii > > jj..

Using patterns, iteration, symmetry, etc. to Using patterns, iteration, symmetry, etc. to define a labeling makes it easier to prove it’s a define a labeling makes it easier to prove it’s a radio labeling.radio labeling.

ninnin

nii

i

xf i

21)(32

13

01

)(

Using an intermediate labelingUsing an intermediate labeling

x11 x0

x9

x10

x12

x13

x14

x15

x16

x1

x2

x5

x6 x4

x3

x8

x7

v3

G8

z

v1

v2

v4

v5

v6

v7

v8

w1

w3

w2

w4 w7

w5

w8

w6

Using patterns, iteration, etc. to prove Using patterns, iteration, etc. to prove the labeling is a radio labelingthe labeling is a radio labeling

For products of cycles and for generalized prism For products of cycles and for generalized prism graphs, the gap was about half the diameter.graphs, the gap was about half the diameter.

This gives | This gives | ff((xxii) – ) – ff((xxjj) | ) | ≥ diam(≥ diam(GG) + 1 whenever ) + 1 whenever jj – – ii ≥ 4. ≥ 4.

Also, Also, | | ff((xxii) – ) – ff((xxjj) | = | ) | = | ff((xxi+ki+k) – ) – ff((xxj+kj+k) |, so it ) |, so it suffices to show the radio condition holds for {suffices to show the radio condition holds for {xx11, , xx22, , xx33, , xx44}.}.

Using symmetry in labeling also advantageous.Using symmetry in labeling also advantageous.

Some Radio Labeling QuestionsSome Radio Labeling Questions Find relationships between Find relationships between rnrn((GG) and specific graph ) and specific graph

properties (e.g. connectivity, diameter, etc.).properties (e.g. connectivity, diameter, etc.).

Investigate radio numbers of various product graphs, Investigate radio numbers of various product graphs, and/or determine the relationship between the radio and/or determine the relationship between the radio number of a product graph and the radio numbers of its number of a product graph and the radio numbers of its factor graphs.factor graphs.

Investigate radio numbers of powers of graphs.Investigate radio numbers of powers of graphs.

Determine properties of minimal labelings. E.g. is the Determine properties of minimal labelings. E.g. is the radio number always realized by a labeling that assigns radio number always realized by a labeling that assigns 1 to a cut vertex? … to a vertex of highest degree?1 to a cut vertex? … to a vertex of highest degree?

Create an algorithm for checking labelings.Create an algorithm for checking labelings.

Find radio numbers of families of graphsFind radio numbers of families of graphs Generalized gears (adapt Generalized gears (adapt

methods)methods) Ladders Ladders Web graphs (products of cycles Web graphs (products of cycles

and paths)and paths) Products of cycles of different Products of cycles of different

sizessizes Grid graphs (products of paths)Grid graphs (products of paths) More generalized prismsMore generalized prisms

LL(3,2,1) labeling(3,2,1) labeling

Clipperton, Gehrtz, Szaniszlo,Clipperton, Gehrtz, Szaniszlo,

and Torkornoo (2006) provide the and Torkornoo (2006) provide the

LL(3,2,1)-labeling numbers for:(3,2,1)-labeling numbers for:

- Complete graphs- Complete graphs - Complete bipartite graphs - Complete bipartite graphs

- Paths- Paths - Cycles - Cycles

- Caterpillars- Caterpillars - - nn-ary trees-ary trees

They also give an upper bound for the They also give an upper bound for the LL(3,2,1)-labeling (3,2,1)-labeling number in terms of the maximum degree of the graph.number in terms of the maximum degree of the graph.

undergraduates

Graceful and Graceful and kk-equitable -equitable labelingslabelings

• Define a labeling Define a labeling ff : : VV((GG) ) → → {0, 1, … |{0, 1, … |EE((GG)|}. )|}.

• Edge (Edge (uu,,vv) receives the label induced by | ) receives the label induced by | ff((uu) – ) – ff((vv) |.) |.

• The labeling is The labeling is gracefulgraceful when none of the vertex or when none of the vertex or edge labels repeat.edge labels repeat.

3

4

1 2

5

0

2 13

54

Graceful and Graceful and kk-equitable -equitable labelingslabelings

• Define a labeling Define a labeling ff : : VV((GG) ) → → {0, 1, … {0, 1, … kk-1}. -1}. • Edge (Edge (uu,,vv) receives the label induced by | ) receives the label induced by | ff((uu) – ) – ff((vv) |.) |.• Let #Let #VVjj and # and #EEjj be the number of vertices and edges, be the number of vertices and edges,

respectively, labeled respectively, labeled jj..• The labeling is The labeling is kk-equitable-equitable if |# if |#VVii - #- #VVjj| | ≤ 1 and ≤ 1 and

|#|#EEii - #- #EEjj| | ≤ 1≤ 1 for all for all ii ≠ ≠ jj in in {0, 1, … {0, 1, … kk-1}.-1}.

0 2 11 2 0k = 3:

2 1 0 1 2

#V0 = #V1 = #V2 = 2 #E0+1 = #E1 = #E2 = 2

What’s known?What’s known?

StarsStars PathsPaths CaterpillarsCaterpillars The Petersen graphThe Petersen graph nn-cycles for -cycles for nn ≡ 0, 3 (4)≡ 0, 3 (4) Symmetric treesSymmetric trees All trees with no more All trees with no more

than four leavesthan four leaves All trees with no more All trees with no more

than 27 verticesthan 27 vertices

StarsStars PathsPaths CaterpillarsCaterpillars Eulerian graphs (conditions)Eulerian graphs (conditions) Cycles (conditions)Cycles (conditions) Wheels (Wheels (kk = 3) = 3) All trees are 3-equitable.All trees are 3-equitable. All trees with fewer than five All trees with fewer than five

leaves are leaves are kk-equit.-equit.

graceful k-equitable

Some Graceful/ Some Graceful/ kk-Equitable Questions-Equitable Questions Investigate particular types of trees to Investigate particular types of trees to

determine whether they are determine whether they are kk-equitable. -equitable. (E.g. complete binary trees are currently (E.g. complete binary trees are currently under investigation.)under investigation.)

Explore whether particular families of Explore whether particular families of graphs are graphs are kk-equitable or graceful.-equitable or graceful.

Investigate whether methods of “gluing” Investigate whether methods of “gluing” graceful trees together to form larger graceful trees together to form larger graceful trees extend to graceful trees extend to kk-equitability.-equitability.

Reading to get startedReading to get started Radio labelingRadio labeling: Chartrand, Erwin, Harary, and Zhang, : Chartrand, Erwin, Harary, and Zhang, Radio Radio

labelings of graphslabelings of graphs, Bull. Inst. Combin. Appl., 33 (2001), 77-, Bull. Inst. Combin. Appl., 33 (2001), 77-85.85.

LL(2,1) labeling(2,1) labeling: Griggs & Yeh, : Griggs & Yeh, Labeling graphs with a Labeling graphs with a condition at distance 2condition at distance 2, SIAM J. Disc. Math., 5 (1992), 586-, SIAM J. Disc. Math., 5 (1992), 586-595.595.

Graceful & Graceful & kk-equitable labeling-equitable labeling: Cahit, : Cahit, Equitable Tree Equitable Tree LabellingsLabellings, Ars. Combin. 40 (1995), 279-286., Ars. Combin. 40 (1995), 279-286.

General surveyGeneral survey: Gallian, : Gallian, A dynamic survey of graph labelingA dynamic survey of graph labeling, , Dynamical Surveys, DS6, Electron. J. Combin. (1998).Dynamical Surveys, DS6, Electron. J. Combin. (1998).

URL for these slides: http://faculty.csuci.edu/cynthia.wyels

ConjecturesConjecturesGraceful labelings were defined in the context of graph Graceful labelings were defined in the context of graph decompositions. decompositions.

Rosa’s Th’m: If a tree Rosa’s Th’m: If a tree TT with with mm edges has a graceful edges has a graceful labeling, then Klabeling, then K22mm+1+1 decomposes into 2 decomposes into 2mm+1 copies of +1 copies of TT. .

(1968)(1968)

Kotzig-Ringel Conj: Every tree has a graceful labeling.Kotzig-Ringel Conj: Every tree has a graceful labeling.

Cahit’s Conj: All trees are Cahit’s Conj: All trees are kk-equitable. (1990)-equitable. (1990)

Are all complete-binary trees Are all complete-binary trees kk--equitable?equitable?

Know true for k = 2^n, n = 0, 1, …, 5. Think method Know true for k = 2^n, n = 0, 1, …, 5. Think method extends to all n.extends to all n.

Know true for k = 2, 3, 4, 5, 6, 7.Know true for k = 2, 3, 4, 5, 6, 7.

Need last step to show true for all k congruent to 0, 1 mod Need last step to show true for all k congruent to 0, 1 mod 4.4.

Why worry about complete binary trees? Would like to Why worry about complete binary trees? Would like to generalize any findings and methods.generalize any findings and methods.

All trees are 3-equitableAll trees are 3-equitableOutline how this was done. Emphasize student Outline how this was done. Emphasize student contributions! contributions!

Rosa’s Th’m: If a tree Rosa’s Th’m: If a tree TT with with mm edges has a graceful edges has a graceful labeling, then Klabeling, then K22mm+1+1 decomposes into 2 decomposes into 2mm+1 copies of +1 copies of TT..

Kotzig-Ringel Conj: Every tree has a graceful labeling.Kotzig-Ringel Conj: Every tree has a graceful labeling.

Cahit’s Conj: All trees are Cahit’s Conj: All trees are kk-equitable.-equitable.