ON -MAGIC HYPERCUBES RINOVIA SIMANJUNTAK rino@math.itb.ac.id - - PowerPoint PPT Presentation

ON 𝐸-MAGIC HYPERCUBES

RINOVIA SIMANJUNTAK

rino@math.itb.ac.id

PALTON ANUWIKSA AKIHIRO MUNEMASA (TOHOKU UNIVERSITY)

COMBINATORIAL MATHEMATICS RESEARCH GROUP FACULTY OF MATHEMATICS AND NATURAL SCIENCES INSTITUT TEKNOLOGI BANDUNG

31st Cumberland Conference, 18-19 May 2019, UCF partially supported by: JSPS Open Partnership Joint Research Project 2017-2019 and RISTEKDIKTI Fundamental Research Grant 2018-2020

Definition Sedláček (1963) A magic labeling is a one-to-one mapping 𝑔: 𝐹→ℝ+ with the property that there is a constant 𝑙 such that at any vertex 𝑦 ෍

𝑧∈𝑂(𝑦)

𝑔 𝑦𝑧 = 𝑙 where 𝑂(𝑦) is the set of vertices adjacent to 𝑦. Definition Kotzig and Rosa (1970) An edge-magic total labeling is a bijection 𝑔: 𝑊 ∪ 𝐹 → {1,2, … , 𝑊 ∪ 𝐹 } with the property that there is a constant k such that at any edge 𝑦𝑧, 𝑔 𝑦 + 𝑔 𝑦𝑧 + 𝑔(𝑧) = 𝑙. A graph admitting edge-magic total labeling is called edge-magic.

Magic Labeling: The Beginning

Not all graphs are edge-magic. The edge-magic property is not monotone with respect to the subgraph relation. Conjecture Kotzig & Rosa (1970) All trees are edge-magic. Question Erdős (Kalamazoo 1996) What is the maximum number of edges ℳ(𝑜) in an edge-magic graph of

• rder 𝑜?

2 7 𝑜2 + 𝑃(𝑜) ≤ ℳ(𝑜) ≤ (0.489 … + 𝑝 1 )𝑜2 Pikhurko (2006)

Magic Labeling: Open Problem and Conjecture

𝐸-Magic Labeling

Definition O’Neal & Slater (2011) Let 𝑒 be the diameter of a graph 𝐻 and 𝐸 ⊆ {0,1,2, … , 𝑒} be a set

• f distances in graph 𝐻.

A bijection 𝑔 ∶ 𝑊→{1, 2, … , 𝑊 } is said to be a 𝐸-magic labeling if there exists a magic constant 𝑙 such that for any vertex 𝑦, the weight 𝑥(𝑦) = σ𝑧∈𝑂𝐸(𝑦) 𝑔 𝑧 = 𝑙, where 𝑂𝐸(𝑦) = {𝑧𝑊 |𝑒(𝑦, 𝑧)  𝐸}. A graph admitting a 𝐸-magic labeling is called 𝐸-magic. If 𝐸 = {1}, a 𝐸-magic labeling is known as a distance magic labeling. Vilfred (1994) If 𝐸 = {0,1}, a 𝐸-magic labeling is called a closed distance magic

• labeling. Beena (2009)

Smallest Distance Magic Connected Graphs

Distance Magic Graphs are Rare

Catalogue of distance magic graphs up to 9 vertices. Yasin & RS (2015)

n # non-isomorphic graphs # non-isomorphic distance magic graphs 1 1 1 2 2 1 3 4 2 4 11 2 5 34 2 6 156 2 7 1044 4 8 12346 6 9 275668 6

Some Observations

 There is no {0}-magic graph.  A regular graph with odd degree is not {1}-magic

(distance magic).

 All graphs are {0,1, … , 𝑒}-magic.  A graph is 𝐸-magic if and only if it is 0,1, … , 𝑒 \𝐸-

magic.

 A graph is {1}-magic (distance magic) if and only if its

complement is {0,1}-magic (closed distance magic).

 Let 𝐸1 and 𝐸2 be two disjoint sets of distances. If a

graph is both 𝐸1-magic and 𝐸2-magic then it is also 𝐸1 ∪ 𝐸2-magic.

The 𝐸-Magic Constant is Unique

Definition O’Neal & Slater (2013) A function g : V → [0, 1] is said to be a 𝐸-neighborhood fractional dominating function if for every vertex v, σ𝑣∈𝑂𝐸(𝑤) 𝑕(𝑤) ≥ 1. The 𝐸-neighborhood fractional domination number of a graph is denoted by 𝛿𝑔𝑢(𝐻; 𝐸) and is defined as

min σ𝑤∈𝑊(𝐻) 𝑕 𝑤 |𝑕 is a D−neigborhood fractional dominating function .

Theorem O’Neal & Slater (2013) If a graph G is 𝐸-magic, then its magic constant is 𝑙 = 𝑜(𝑜 + 1) 2𝛿𝑔𝑢(𝐻; 𝐸)

Distance Regular Graphs

Let 𝐻𝑗 𝑣 denote the set of vertices at distance 𝑗 from 𝑣. A connected graph 𝐻 of diameter 𝑒 is distance-regular if it is 𝑠- regular and there exist positive integers 𝑐0 = 𝑠, 𝑐1, … , 𝑐𝑒−1, 𝑑1 = 1, 𝑑2, … , 𝑑𝑒, such that for every pair of vertices 𝑣 and 𝑤 where 𝑒 𝑣, 𝑤 = 𝑘, there are exactly 𝑑

𝑘 neighbors of 𝑣 in 𝐻 𝑘−1 𝑤 and exactly 𝑐 𝑘 neighbors of 𝑣

in 𝐻

𝑘+1 𝑤 .

The array 𝑠, 𝑐1, … , 𝑐𝑒−1; 1, 𝑑2,…,𝑑𝑒 is called the intersection array of 𝐻.

𝑘 +1 𝑤 𝑣 𝑘 𝑘 − 1 𝑑

𝑘

𝑐

𝑘

𝐸-Magic Strongly-Regular Graphs

Theorem Anholcer, S. Cichacz, and I. Peterin (2016) RS and Anuwiksa (2018+) The only 𝐸-magic strongly-regular graphs are

1)

the complete multipartite graphs 𝐼𝑜−𝑠, 𝑜

𝑜−𝑠 where

𝑜 − 𝑠 is even or both 𝑜 − 𝑠 and

𝑜 𝑜−𝑠 are odd, for

𝐸 = {1} or 𝐸 = {0,2},

2)

the complete graphs 𝐿𝑜, for 𝐸 = {0,1} or 𝐸 = {2}.

{1}-Magic Distance Regular Graphs

• f Diameter 3

A distance-𝑙 graph of a graph 𝐻 is a graph with the same vertex set as 𝐻 and edge set consisting of the pairs of vertices that lie a distance 𝑙 apart. A distance-regular graph is primitive if all of its distance-𝑙 graphs are connected. Theorem RS and Anuwiksa (2018+) Let 𝐻 be a distance-regular graph of diameter 3. If 𝐻 is distance magic then 𝐻 is primitive. Note that in the BCN table [Brouwer, Cohen, Neumaier (1989)] of distance-regular graphs on at most 4096 vertices, there are 4 feasible primitive graphs: Perkel graph (on 57 vertices), unitary non-isotropic graph (525), Moscow-Soicher graph (672), and Brouwer graph (729).

The Tridiagonal Matrix of a Distance Regular Graph

Theorem Biggs (1996) If 𝐻 is a distance-regular graph of diameter 𝑒 and intersection array 𝑙, 𝑐1, … , 𝑐𝑒−1; 1, 𝑑2,…,𝑑𝑒 , then 𝐻 has 𝑒+1 distinct eigenvalues which are the eigenvalues of the tridiagonal (𝑒 + 1) × (𝑒 + 1) matrix 𝐶 = 1 𝑠 𝑏1 ⋯ 𝑑2 𝑐1 𝑏2 𝑑3 𝑐2 𝑏3 ⋮ ⋱ ⋮ ⋱ ⋯ ⋱ 𝑑𝑒 𝑐𝑒−1 𝑏𝑒

The Tridiagonal Matrix

Let 𝐻 be a distance-regular graph of diameter 𝑒. For a vertex 𝑦 and a labeling of vertices 𝑚, 𝑇𝑗 𝑦 = σ𝑧∈𝐻𝑗(𝑦) 𝑚(𝑧), the sum of labels of all vertices in 𝐻𝑗 𝑦 . It is clear that 𝑇0 𝑦 = 𝑚(𝑦). The vector 𝒕(𝒚) is 𝑇𝑗 𝑦

𝑗=0 𝑒 .

The vector 𝒍 is |𝐻𝑗 𝑦 | 𝑗=0

𝑒

. We suppress the reference to 𝑦 since |𝐻𝑗 𝑦 | is independent of the choice of a vertex 𝑦. Lemma If 𝑚 is a distance magic labeling with magic constant 𝑙′, then 𝐶𝒕 𝒚 = 𝑙′𝒍. If 𝑚 is a closed distance magic labeling with magic constant 𝑙′, then (𝐽 + 𝐶)𝒕 𝒚 = 𝑙′𝒍.

Antipodal Double Cover

A distance-regular graph 𝐻 of diameter 𝑒 is called an antipodal double cover if 𝐻𝑒(𝑦) = 1 for some (and hence all) 𝑦 ∈ 𝑊(𝐻). The unique vertex in 𝐻𝑒(𝑦) is called the antipode of 𝑦 and denoted by 𝑦′. Suppose that 𝑚 is a distance magic labeling of 𝐻 with magic constant 𝑙′. For a vertex 𝑦 in 𝐻, we have 𝑙′𝒍 = 𝐶𝒕 𝒚 = 𝐶𝒕 𝒚′ =

𝑙′ 𝒔 𝐶𝒍.

Thus 𝒕 𝒚 − 𝒕 𝒚′ ∈ Ker 𝐶, which means Ker 𝐶 has a basis of the form 1 ⋮ −1 . And so 𝑚 𝑦 + 𝑚 𝑦′ is a constant independent of 𝑦. Consequently, for every 𝑦, 𝑇

𝑘 𝑦 + 𝑇𝑒−𝑘 𝑦 = σ𝑧∈𝐻𝑘(𝑦) 𝑚 𝑧 + 𝑚 𝑧′ is a

constant. Theorem Let 𝐻 be a distance-regular graph of diameter 𝑒 which is an antipodal double cover. If 𝑚 is a distance magic labeling or a closed distance magic labeling of 𝐻 then 𝑚 is also {𝑘, 𝑒 − 𝑘} − magic for all 𝑘.

Bipartite Antipodal Double Cover

If 𝐻 is bipartite, then Ker 𝐶 ≠ 0 implies that 𝑒 is even. Recursively comparing entries of 𝐶𝒕 𝒚 = 𝑙′𝒍, there exist constants 𝑏 and 𝑐 such that 𝑚(𝑦′) = 𝑏 + 𝑐𝑚(𝑦). More explicitly, 𝑐 = (−1)𝑒/2. Switching the role of 𝑦 and 𝑦′, 𝑚(𝑦) = 𝑏 + 𝑐𝑚(𝑦′). This forces 𝑐 = −1, and 𝑒 ≡ 2 𝑛𝑝𝑒4 . Theorem Let 𝐻 be a bipartite distance-regular graph which is an antipodal double cover with diameter 𝑒. If 𝐻 has distance magic labeling then 𝑒 ≡ 2 𝑛𝑝𝑒4 . Corollaries

 Hadamard graphs are not distance magic.  For 𝑜 ≢ 2(𝑛𝑝𝑒4), the hypercube 𝑅𝑜 is not distance magic. Cichacz,

Froncek, Krop, Raridan (2016) Theorem Gregor and Kovář (2013) The hypercube 𝑅𝑜 is distance magic if and only if 𝑜 ≡ 2(𝑛𝑝𝑒4).

Many Labelings from One

Theorem Let 𝐻 be a distance-regular graph of diameter 𝑒 and let 𝐸 ⊆ {0,1,2, … , 𝑒} be a non-empty set of distances. If 𝐻 admits a distance magic labeling 𝑚 then 𝑚 is either 𝐸-magic or (𝛽, 𝜀) − 𝐸- antimagic for some 𝛽, 𝜀. Morever, if 𝐻 is bipartite, then 𝑚 is 𝐸-magic for all non-empty 𝐸 ⊆ {1,3,5, … }. Theorem If 𝑜 ≡ 2(𝑛𝑝𝑒4) then there exists a 𝐸-magic labeling of the hypercube 𝑅𝑜 whenever 𝐸 is of the form 𝐹 ∪ ራ

𝑗∈𝐽

𝑗, 𝑜 − 𝑗 , where 𝐹 ⊆ 1,3,5, … , 𝐽 ⊆ {0,1, … ,

𝑜 2}, and 𝐹 ∩ 𝑗, 𝑜 − 𝑗 = ∅ (𝑗 ∈ 𝐽).

Open Problem Are these the only 𝐸s for which the hypercube 𝑅𝑜, 𝑜 ≡ 2(𝑛𝑝𝑒4), is 𝐸- magic?

More Labelings for Hypercubes

For a vector 𝒃 = (𝑏0, … , 𝑏𝑜−1) ∈ 𝔾2

𝑜, 𝜂 𝒃 = σ𝑗=0 𝑜−1 𝑏𝑗2𝑗.

A subset 𝐵 ⊆ 𝔾2

𝑜 is balanced if 𝒃 ∈ 𝐵|𝑏𝑗 = 1

=

𝐵 2

∀𝑗 ∈ 1, … , 𝑜 . For a distance set 𝐸 ⊆ 0,1, … , 𝑜 , a bijection 𝑔: 𝔾2

𝑜 → 𝔾2 𝑜 is 𝐸-neighbor balanced if

ڂ𝑗∈𝐸 𝑔 𝐻𝑗(𝒗) is balanced for every vertex 𝒗 of 𝑅𝑜. Lemmas



If 𝐵 is a balanced subset of 𝔾2

𝑜 then σ𝒃∈𝐵 𝜂 𝒃 = |𝐵| 2

2𝑜 − 1 .



If 𝑔 is 𝐸-neighbor balanced bijection then 𝜂 ∘ 𝑔 is a 𝐸-magic labeling of 𝑅𝑜.



Let 𝑔: 𝔾2

𝑜 → 𝔾2 𝑜 be a nonsingular linear transformation. 𝑔 is closed neighbor-balanced if and

• nly if the matrix representation of 𝑔 with respect to the standard basis has constant row sum

𝑜+1 2 .



For 𝑜 = 4𝑛 + 1, the matrix 𝑁 = 1 12𝑛 𝐽2𝑛 𝐾2𝑛 𝐾2𝑛 𝐽2𝑛 is non-singular and has constant row sum

𝑜+1 2 .

Theorem The hypercube 𝑅𝑜 is closed distance magic if and only if 𝑜 ≡ 1(𝑛𝑝𝑒4).

More Labelings for Hypercubes

For an integer 𝑗 (0 ≤ 𝑗 ≤ 𝑒), 𝐵𝑗 is the 𝑗-distance matrix of 𝐻. Lemma

• i. If 𝑜 is odd then Ker(𝐵0 + 𝐵1) ⊆ Ker(𝐵2𝑗 + 𝐵2𝑗+1) (𝑗 = 0, 1, … ,

𝑜−1 2 ),

• ii. If 𝑜 ≡ 1(𝑛𝑝𝑒4) then Ker(𝐵0 + 𝐵1) ⊆ Ker(𝐵𝑗 + 𝐵𝑜−𝑗) (𝑗 = 0, 1, … ,

𝑜−1 2 ).

Theorem Let 𝑜 ≡ 1(𝑛𝑝𝑒4). The hypercube 𝑅𝑜 is 𝐸-distance magic, whenever 𝐸 is of the form ራ

𝑗∈𝐽1

{2𝑗, 2𝑗 + 1} ∪ ራ

𝑘∈𝐽2

𝑘, 𝑜 − 𝑘 , where 𝐽1, 𝐽2 ⊆ {0,1, … ,

𝑜−1 2 } and 2𝑗, 2𝑗 + 1 ∩ 𝑘, 𝑜 − 𝑘 = ∅ (𝑗 ∈ 𝐽1, 𝑘 ∈ 𝐽2).

Open Problem Are these the only 𝐸 for which the hypercube 𝑅𝑜, 𝑜 ≡ 1(𝑛𝑝𝑒4), is 𝐸-magic?

Open Problem

For 𝑜 ≡ 0,3 (𝑛𝑝𝑒4), does there exist a non- empty distance set 𝐸 ⊂ {0,1,2, … , 𝑒} for which the hypercube 𝑅𝑜 is 𝐸-magic? In general, Let 𝐻 be a distance-regular graph of diameter 𝑒. Does there always exist a non-empty distance set 𝐸 ⊂ {0,1,2, … , 𝑒} such that 𝐻 is 𝐸-magic?

TERIMA KASIH