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 31 st 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

Magic Labeling: The Beginning 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 ෍ 𝑔 𝑦𝑧 = 𝑙 𝑧∈𝑂(𝑦) 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: Open Problem and Conjecture 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 order 𝑜 ? 7 𝑜 2 + 𝑃(𝑜) ≤ ℳ(𝑜) ≤ (0.489 … + 𝑝 1 )𝑜 2 Pikhurko (2006) 2

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 of 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) = σ 𝑧∈𝑂 𝐸 (𝑦) 𝑔 𝑧 = 𝑙 , where 𝑂 𝐸 (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) # non-isomorphic graphs # non-isomorphic n 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 𝐸 1 and 𝐸 2 be two disjoint sets of distances. If a graph is both 𝐸 1 -magic and 𝐸 2 -magic then it is also 𝐸 1 ∪ 𝐸 2 -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, σ 𝑣∈𝑂 𝐸 (𝑤) 𝑕(𝑤) ≥ 1 . The 𝐸 -neighborhood fractional domination number of a graph is denoted by 𝛿 𝑔𝑢 (𝐻; 𝐸) and is defined as min σ 𝑤∈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) 2𝛿 𝑔𝑢 (𝐻; 𝐸)

Distance Regular Graphs Let 𝐻 𝑗 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 𝑐 0 = 𝑠 , 𝑐 1 , … , 𝑐 𝑒−1 , 𝑑 1 = 1 , 𝑑 2 , … , 𝑑 𝑒 , 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 . 𝑑 𝑘 𝑣 𝑘 − 1 𝑘 𝑐 𝑘 + 1 𝑘 𝑤 The array 𝑠, 𝑐 1 , … , 𝑐 𝑒−1 ; 1, 𝑑 2 ,…, 𝑑 𝑒 is called the intersection array of 𝐻 .

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 the complete multipartite graphs 𝐼 𝑜−𝑠, 𝑜 𝑜−𝑠 where 1) 𝑜 𝑜 − 𝑠 is even or both 𝑜 − 𝑠 and 𝑜−𝑠 are odd, for 𝐸 = {1} or 𝐸 = {0,2} , the complete graphs 𝐿 𝑜 , for 𝐸 = {0,1} or 𝐸 = 2) {2} .

{1} -Magic Distance Regular Graphs of 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, 𝑐 1 , … , 𝑐 𝑒−1 ; 1, 𝑑 2 ,…, 𝑑 𝑒 , then 𝐻 has 𝑒 +1 distinct eigenvalues which are the eigenvalues of the tridiagonal (𝑒 + 1) × (𝑒 + 1) matrix 0 1 0 ⋯ 0 𝑠 𝑏 1 𝑑 2 𝑏 2 𝑑 3 0 𝑐 1 ⋮ 𝐶 = 𝑐 2 𝑏 3 ⋱ ⋱ 𝑑 𝑒 ⋮ ⋱ 𝑐 𝑒−1 𝑏 𝑒 0 ⋯

The Tridiagonal Matrix Let 𝐻 be a distance-regular graph of diameter 𝑒 . For a vertex 𝑦 and a labeling of vertices 𝑚 , 𝑇 𝑗 𝑦 = σ 𝑧∈𝐻 𝑗 (𝑦) 𝑚(𝑧) , the sum of labels of all vertices in 𝐻 𝑗 𝑦 . It is clear that 𝑇 0 𝑦 = 𝑚(𝑦) . 𝑒 . The vector 𝒕(𝒚) is 𝑇 𝑗 𝑦 𝑗=0 𝑒 The vector 𝒍 is |𝐻 𝑗 𝑦 | 𝑗=0 . We suppress the reference to 𝑦 since |𝐻 𝑗 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 𝐻 𝑒 (𝑦) = 1 for some (and hence all) 𝑦 ∈ 𝑊(𝐻) . The unique vertex in 𝐻 𝑒 (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′ . 𝑙 ′ For a vertex 𝑦 in 𝐻 , we have 𝑙 ′ 𝒍 = 𝐶𝒕 𝒚 = 𝐶𝒕 𝒚′ = 𝒔 𝐶𝒍 . Thus 𝒕 𝒚 − 𝒕 𝒚 ′ ∈ Ker 𝐶 , which means Ker 𝐶 has a basis of the form 1 . And so 𝑚 𝑦 + 𝑚 𝑦 ′ is a constant independent of 𝑦 . ⋮ −1 Consequently, for every 𝑦 , 𝑇 𝑘 𝑦 + 𝑇 𝑒−𝑘 𝑦 = σ 𝑧∈𝐻 𝑘 (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 𝑅 𝑜 is not distance magic. Cichacz, Froncek, Krop, Raridan (2016) Theorem Gregor and Kovář (2013) The hypercube 𝑅 𝑜 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 𝑅 𝑜 whenever 𝐸 is of the form 𝐹 ∪ ራ 𝑗, 𝑜 − 𝑗 , 𝑗∈𝐽 𝑜 where 𝐹 ⊆ 1,3,5, … , 𝐽 ⊆ {0,1, … , 2 } , and 𝐹 ∩ 𝑗, 𝑜 − 𝑗 = ∅ (𝑗 ∈ 𝐽) . Open Problem Are these the only 𝐸 s for which the hypercube 𝑅 𝑜 , 𝑜 ≡ 2(𝑛𝑝𝑒4) , is 𝐸 - magic?