Partially Magic Labelings The AntimagicGraph Conjecturemaryamf/poster.pdf · Computing the...
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−Labeling: each edge label is among 0,1, ⋯ , ) ∶ the number of magic labelings of of index Theorem (Stanley 1973): For a finite graph , there exist polynomials ) () and ) () such that ) = ) + −1 4 ) (). Theorem (Stanley 1973): If the graph G minus its loops is bipartite , then ) () is a polynomial of . Labeling: of a graph is an assignment : → ℤ ;< of a nonnegative integer to each edge of Partially Magic Labelings & Magic Magic Labeling: • each edge label 0,1, ⋯ , || is used exactly once • the sums of the labels on all edges incident with a given node are equal to Example of Magic labeling on @ of index 19 11 7 1 8 4 5 3 2 6 10 19 19 19 19 19 Antimagic Labeling: • each edge label 0,1, ⋯ , || is used exactly once • the sum of the labels on all edges incident with a given node is unique Antimagic 18 23 24 22 20 2 3 9 7 6 8 10 4 5 1 Antimagic Graph Conjecture Every connected graph except for A admits an antimagic labeling. Surprise: still open for trees ! Proved for: • trees without vertices of degree 2 • connected graphs with minimum degree ≥ log || • −regular graphs J : the number of partially magic −labelings of over Theorem: J is a quasi- polynomial in with period at most 2. Theorem: If the graph G minus its loops is bipartite, then J () is a polynomial in . Partially magic labeling of over ⊆ : is a labeling such that “magic occurs” just in , that is, the sums of the labels on all edges incident with a given node in are equal The Antimagic Graph Conjecture References [1] R. Stanley. Linear homogeneous Diophantine equations and magic labelings of graphs. Duke Math. J., 40:607–632, 1973. [2] J. Gallian. A dynamic survey of graph labeling. Electron. J. Combin., 5:Dynamic Survey 6, 43 pp. (electronic), 1998. [3] Nora Hartsfield and Gerhard Ringel. Pearls in Graph Theory: A Comprehensive Introduction. Dover Publications, Inc., Mineola, NY, revised edition, 2003. [4] M. Beck and S. Robins. Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, New York, second edition, 2015. Partially Magic Generalization Application Theorem: ) is a quasi-polynomial in of period at most 2. Theorem: If the graph minus its loops is bipartite , then ) is a polynomial in . Main Theorem ) : the number of weakly antimagic −labelings Example of , : Weakly antimagic 3-labelings, edge labels ∈ 0,1,2,3 3 0 2 1 4 0 3 1 5 0 3 2 6 1 3 2 T U,V 3 =4 T U,V = +1 3 Polynomial in 4 3 2 2 0 1 S 7 Matthias Beck 7 1 2 Motivation Quasi-polynomial of period at most 2 Every bipartite graph without a A -component admits a weakly antimagic labeling. Corollary Birkhoff [1912] introduced the chromatic polynomial ) (the number of proper -colorings of a graph) in an attempt to prove the Four Color Theorem: ) 4 > 0 if is planar. Our idea: ) : the number of weakly antimagic k-labelings • G has a weak antimagic labeling ) >0 • ) = ∑ J J () J⊆] |J|;A for some integers J • Why weakly? The counting function of antimagic -labelings is in general not a polynomial! • Directed Antimagic Graph Conjecture • Distinct Antimagic Counting Maryam Farahmand Open Problems: Berkeley UNIVERSITY OF CALIFORNIA 7
Transcript of Partially Magic Labelings The AntimagicGraph Conjecturemaryamf/poster.pdf · Computing the...
𝒌 −Labeling: each edge label is among 0,1,⋯ , 𝑘
𝐻) 𝑟 ∶the number of magic labelings of 𝐺 ofindex 𝑟
Theorem (Stanley 1973): For a finite graph𝐺, there exist polynomials 𝑃)(𝑟) and𝑄)(𝑟)such that 𝐻) 𝑟 = 𝑃) 𝑟 + −1 4𝑄)(𝑟).
Theorem (Stanley 1973): If the graph G minusits loops is bipartite, then 𝐻)(𝑟)is a polynomialof 𝑟.
Labeling: of a graph 𝐺 is an assignment 𝐿: 𝐸 → ℤ;<of a nonnegative integer 𝐿 𝑒 to each edge 𝑒 of 𝐺
Partially Magic Labelings &
Magic
Magic Labeling:• each edge label 0,1,⋯ , |𝐸| is used
exactly once• the sums of the labels on all edges incident
with a given node are equal to 𝑟
Example of Magic labeling on 𝑊@of index 19
11
7
1
8
4
5
3
2
6
10
19
19
1919
19
Antimagic Labeling:• each edge label 0,1,⋯ , |𝐸| is used exactly
once• the sum of the labels on all edges incident
with a given node is unique
Antimagic
18
23
2422
20
2 3
976
8 10 4
5
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Antimagic Graph Conjecture
Every connected graph except for 𝑘Aadmits an antimagic labeling.
Surprise: still open for trees !
Proved for: • trees without vertices of degree 2• connected graphs with minimum
degree ≥ 𝑐 log |𝑉|• 𝑘 −regular graphs
𝑀J 𝑘 : the number of partially magic𝑘 −labelings of 𝐺over 𝑆
Theorem: 𝑀J 𝑘 is a quasi- polynomial in𝑘with period at most 2.
Theorem: If the graph G minus its loops isbipartite, then 𝑀J(𝑘) is a polynomial in 𝑘.
Partially magic labeling of 𝐺 over 𝑆𝑆 ⊆ 𝑉 𝐺 : is a labeling such that “magic
occurs” just in 𝑆, that is, the sums of the labels on all edges incident with a given node in 𝑆 are equal
The Antimagic Graph Conjecture
References[1] R. Stanley. Linear homogeneous Diophantine equations and magic labelings of graphs. Duke Math. J., 40:607–632, 1973.
[2] J. Gallian. A dynamic survey of graph labeling. Electron. J. Combin., 5:Dynamic Survey 6, 43 pp. (electronic), 1998.
[3] Nora Hartsfield and Gerhard Ringel. Pearls in Graph Theory: A Comprehensive Introduction. Dover Publications, Inc., Mineola, NY, revised edition, 2003.
[4] M. Beck and S. Robins. Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, New York, second edition, 2015.
Partially Magic
Generalization
Application
Theorem: 𝐴) 𝑘 is a quasi-polynomial in 𝑘of period at most 2.
Theorem: If the graph 𝐺 minus its loops is bipartite, then 𝐴) 𝑘 is a polynomial in 𝑘.
Main Theorem
𝐴) 𝑘 : the number of weakly antimagic𝑘 −labelings
Example of 𝑨𝑲𝟑,𝟏 𝑘 :
Weakly antimagic 3-labelings, edge labels ∈ 0,1,2,3
3
0
2
14
0
3
15
0
3
26
1
3
2 𝐴TU,V 3 = 4
𝐴TU,V 𝑘 = 𝑘 + 13
Polynomial in 𝑘4
3
2
2
0
1S
7
Matthias Beck
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Motivation
Quasi-polynomial of period at most 2
Every bipartite graph without a 𝐾A-component admits a weakly antimagic labeling.
Corollary
Birkhoff [1912] introduced the chromatic polynomial 𝜒) 𝑘(the number of proper 𝑘-colorings of a graph) in an attempt to prove the Four Color Theorem: 𝜒) 4 > 0 if 𝐺 is planar.
Our idea: 𝐴) 𝑘 : the number of weakly antimagic k-labelings
• G has a weak antimagic labeling 𝐴) 𝐸 > 0
• 𝐴) 𝑘 = ∑ 𝑐J𝑀J(𝑘)�J⊆]|J|;A
for some integers𝐶J
• Why weakly? The counting function of antimagic𝑘-labelings is in general not a polynomial!
• Directed Antimagic Graph Conjecture • Distinct Antimagic Counting
Maryam Farahmand
Open Problems:
BerkeleyUNIVERSITYOFCALIFORNIA
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