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

1

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

7 1

2

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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