1. Shifted-antimagic labelings for graphs
Zhishi Pan
Department of Mathematics
Tamkang University
Joint works:Fei-Huang Chang, Hong-Bin Chen and Wei-Tian Li
2019/8/20

2. Magic 1 2 3 4 5 6 7 8 9
Magic square
河圖洛書

3. Magic labeling of graph
Let 𝐺=(𝑉,𝐸) is a simple graph
[Sedláček, 1963]
A magic labeling is a function 𝑓
𝑓:𝐸 𝐺 → 𝑅 + ∪{0} s.t. ∀𝑢,𝑣∈𝑉 𝐺 , 𝑢≠𝑣, 𝑒∈𝐸(𝑢) 𝑓(𝑒 )= 𝑒∈𝐸(𝑣) 𝑓(𝑒)
[Stewart, 1966]
A magic labeling supermagic if the set of edge labels consisted of consecutive integers.
𝐾 𝑛 is supermagic for 𝑛≥5 ⟺ 𝑛>5 and 𝑛≡0 mod 4.
Ex: 𝐾 6 1 3 2 4 5 6 7 9 8 11 10 13 12 15 14

4. Antimagic labeling [Hartsfield, Ringel, 1990]
An anti-magic labeling of graph is a bijection 𝑓
𝑓:𝐸 𝐺 →{1,2,…, 𝐸 } s.t. ∀𝑢,𝑣∈𝑉 𝐺 , 𝑢≠𝑣, 𝑒∈𝐸(𝑢) 𝑓(𝑒 )≠ 𝑒∈𝐸 𝑣 𝑓(𝑒 ).
A graph 𝐺 is called antimagic if 𝐺 admits an antimagic labeling.
𝑃 𝑛 (𝑛≥3), 𝐶 𝑛 , 𝑊 𝑛 and 𝐾 𝑛 (𝑛≥3) are antimagic.
Ex: 1 1 2 3 4 5 6 1 2 3 4 5 6 10 12 3 2 3 4 8 11 10 6 1
𝐶 6
𝑊 6
𝐾 6 3 1 6 5 4 2 7 8 5 3 15 9 10 18 7 8 4 11 12 15 14 13 12 11 7 8 9 10 9 5 5 4 21 23
𝑃 6

5. Antimagic labeling [Hartsfield, Ringel 1990]
Conjecture: Every graph except for 𝐾 2 are anti-magic.
[Hartsfield, Ringel 1990]
Conjecture: Every tree except for 𝐾 2 are anti-magic.
[Alon, Kaplan, Lev, Roditty, Yuster, 2004]
All graphs with 𝑛(≥4) vertices and minimum degree Ω log 𝑛 are anti-magic.
[Alon, Kaplan, Lev, Roditty, Yuster, 2004]
If 𝐺 is a graph with 𝑛(≥4) vertices and the maximum degree Δ 𝐺 ≥𝑛−2,
then 𝐺 is anti-magic.
[Alon, Kaplan, Lev, Roditty, Yuster, 2004]
All complete partite graphs except 𝐾 2 are antimagic.

6. Antimagic labeling [Hefetz, 2005]
A graph 𝐺 with 3 𝑘 vertices is antimagic if it admits a 𝐾 3 -factor.
[Hefetz, Saluz, Tran, 2010]
A graph with 𝑝 𝑚 vertices, where 𝑝 is an odd prime and 𝑚 is positive, and a
𝐶 𝑝 factor is antimagic.
[Wang, 2005]
If 𝐺 is an 𝑟-regular antimagic graph with 𝑟>1 then 𝐺× 𝐶 𝑛 is anti-magic.
[Wang, 2005]
The toroidal grids 𝐶 𝑛 1 × 𝐶 𝑛 2 ×…× 𝐶 𝑛 𝑘 are antimagic.
[Cheng, 2008]
All Cartesian products or two or more regular graphs of positive degree are
antimagic.
[Cheng, 2007]
𝑃 𝑛 1 × 𝑃 𝑛 2 ×…× 𝑃 𝑛 𝑡 (𝑡>2) and 𝐶 𝑚 × 𝑃 𝑛 are antimagic.

7. Antimagic labeling [Wang, Hsiao, 2008]
The following graphs are anti-magic: 𝐺× 𝑃 𝑛 (𝑛>1) and 𝐺× 𝐾 1,𝑛 where 𝐺
is regular; compositions 𝐺[𝐻] where 𝐻 is 𝑑-regular with 𝑑>1; and the
Cartesian product of any double star and a regular graph.
[Solairaju, Arockiasamy, 2010]
Various families of subgraphs of grids 𝑃 𝑚 × 𝑃 𝑛 are antimagic.
[Liang, Zhu, 2013]
If 𝐺 is 𝑘-regular (𝑘>2), then for any graph 𝐻 with 𝐸 𝐻 ≥ 𝑉 𝐻 −1≥1,
the Cartesian product 𝐻×𝐺 is antimagic.
[Liang, Zhu, 2013]
If 𝐸 𝐻 ≥ 𝑉 𝐻 −1 and each connected component of 𝐻 has a vertex of
odd degree, or 𝐻 has at least 2 𝑉 𝐻 −2 edges, then the prism of 𝐻 is
antimagic.

8. Antimagic labeling [Yilma, 2013]
A connected graph with Δ 𝐺 ≥ 𝑉 𝐺 −3,|𝑉 𝐺 |≥9 is antimagic and
that if 𝐺 is a graph with Δ 𝐺 = deg 𝑢 = 𝑉 𝐺 −𝑘, where 𝑘≤|𝑉 𝐺 |/3
and there exists a vertex 𝑣 in 𝐺 such that the union of neighborhoods of the
vertices 𝑢 and 𝑣 forms the whole vertex set 𝑉(𝐺), then 𝐺 is antimagic.
[Cranston, 2009]
For 𝑘≥2, every 𝑘-regular bipartite graph is antimagic.
[Liang, Zhu, 2014]
Every cubic graph is antimagic.
[Cranston, Liang, Zhu, 2015]
Odd degree regular graphs are antimagic.
[Chang, Liang, Pan, Zhu, 2016]
Every even degree regular graph is antimagic.

9. 𝑘-shifted-antimagic labeling
Let 𝐺 be a graph with |𝐸 𝐺 |=𝑚. Given 𝑘∈𝑍, if there exists an injective function 𝑓 from 𝐸(𝐺) to {𝑘+1, 𝑘+2, …, 𝑘+𝑚} such that the vertex sums 𝜙 𝑓 (𝑣) are all distinct for all vertices 𝑣∈𝑉(𝐺), then we say 𝐺 is 𝑘-shifted-antimagic and 𝑓 is a 𝑘-shifted-antimagic labeling of 𝐺. An antimagic graph is 0-shifted antimagic.
Ex: 1 3 5 7 9 2 5 7 6 9 11
𝑃 6 1 2 5 4 3 6 2 5 4 3
1-shifted-antimagic labeling

10. 𝑘-shifted-antimagic labeling
Let 𝐺 be a graph with |𝐸 𝐺 |=𝑚. Given 𝑘∈𝑍, if there exists an injective function 𝑓 from 𝐸(𝐺) to {𝑘+1, 𝑘+2, …, 𝑘+𝑚} such that the vertex sums 𝜑 𝑓 (𝑣) are all distinct for all vertices 𝑣∈𝑉(𝐺), then we say 𝐺 is 𝑘-shifted-antimagic and 𝑓 is a 𝑘-shifted-antimagic labeling of 𝐺. An antimagic graph is 0-shifted antimagic.
Problem: Is it true that for any graph other than 𝐾 2 is 𝑘-shifted-antimagic, for some a positive integer 𝑘?
Problem: Is it true that for any tree other than 𝐾 2 is 𝑘-shifted-antimagic, for some a positive integer 𝑘?
[Chang, Chen, Li, Pan, 2018]
Every tree except for 𝐾 2 is 𝑘-shifted-antimagic for some 𝑘∈𝑁.
[Chang, Chen, Li, Pan, 2018]
Every graph consists of vertices of odd degrees except for 𝐾 2 is 𝑘-shifted-
antimagic for some 𝑘∈𝑁.

11. 𝑘-shifted-antimagic labeling
[Chang, Chen, Li, Pan, 2018]
Every tree except for 𝐾 2 is 𝑘-shifted-antimagic for some 𝑘∈𝑁.
Lemma: Given a graph 𝐺, if there exists an injective function 𝑓:𝐸 𝐺 →{1,2,…,𝑚}, s.t. 𝜙 𝑓 𝑣 ≠ 𝜙 𝑓 (𝑢) whenever deg (𝑣) = deg (𝑢) , for 𝑣≠𝑢, then 𝐺 is 𝑘-shifted-antimagic for any sufficiently large 𝑘. 21 11 33 12 17 18 19 20 21 52 26 27 28 29 30 31 54
101 1 2 3 4 5 8 10 9 11 12 13 14 1 15 16 54 27 22 23 24 25 44 42 81 32 33 34 35 65 58
119 63
134 23 8 9 2 12 16 6 9 6 3 13 7 4 5

12. 𝑘-shifted-antimagic labeling
[Chang, Chen, Li, Pan, 2018]
Every tree except for 𝐾 2 is 𝑘-shifted-antimagic for some 𝑘∈𝑁.
Lemma: Given a graph 𝐺, if there exists an injective function 𝑓:𝐸 𝐺 →{1,2,…,𝑚}, s.t. 𝜙 𝑓 𝑣 ≠ 𝜙 𝑓 (𝑢) whenever deg (𝑣) = deg (𝑢) , for 𝑣≠𝑢, then 𝐺 is 𝑘-shifted-antimagic for any sufficiently large 𝑘. 2 3 4 5 6 16 19 9 11 10 12 13 14 15 23 36 17 58 29 18 20 21 22 55 24 25 26 46 44 85 27 28 30 31 32 56
105 33 34 35 67 60
123 65
138 2 9 3 7 4 26 8 5 10 6

13. 𝑘-shifted-antimagic labeling
[Chang, Chen, Li, Pan, 2018]
Every tree except for 𝐾 2 is 𝑘-shifted-antimagic for some 𝑘∈𝑁.
Lemma: Given a graph 𝐺, if there exists an injective function 𝑓:𝐸 𝐺 →{1,2,…,𝑚}, s.t. 𝜙 𝑓 𝑣 ≠ 𝜙 𝑓 (𝑢) whenever deg (𝑣) = deg (𝑢) , for 𝑣≠𝑢, then 𝐺 is 𝑘-shifted-antimagic for any sufficiently large 𝑘.
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14. 𝑘-shifted-antimagic labeling
[Chang, Chen, Li, Pan, 2018]
Every graph consists of vertices of odd degrees except for 𝐾 2 is 𝑘-shifted-
antimagic for some 𝑘∈𝑁.
Sketch proof: ≤ ≤ ≤ ≤

15. 𝑘-shifted-antimagic labeling
An antimagic labeling 𝑓 satisfying 𝜙 𝑓 (𝑢) < 𝜙 𝑓 (𝑣) whenever deg 𝑢 < deg 𝑣 , is called strongly antimagic.
If 𝐺 is a strongly antimagic graph, then it is 𝑘-shifted-antimagic for all 𝑘≥0.
Consider for 𝑘<0
If 𝐺 is a 𝑘-shifted-antimagic graph , then it is −(𝑚+𝑘+1)-shifted-antimagic.
If 𝐺 is a 𝑘-shifted-antimagic graph for all integer 𝑘, then 𝐺 is called absolutely antimagic.
Every path 𝑃 𝑛 with 𝑛≥6 is absolutely antimagic.

16. Graphs that are not 𝑘−shifted−antimagic
Trees with diameter at most four
For 𝑛≥2, the star 𝑆 𝑛 is 𝑘-shifted-antimagic if and only if 𝑘∉{− 𝑛 2 −1 ,− 𝑛 2 }
A double star 𝑆 𝑎,𝑏 is absolutely antimagic if and only if 𝑎,𝑏 ∉{(2,1)}∪ {(2𝑖−1,1) | 𝑖≥1}. Moreover, 𝑆 2,1 is 𝑘-shifted-antimagic if and only if 𝑘∉ {−2,−3}, and 𝑆 2𝑖−1,1 is 𝑘-shifted-antimagic if and only if 𝑘≠−(𝑖+1).
The path 𝑃 5 is 𝑘-shifted-antimagic if and only if 𝑘∉{−2,−3}.

17. Graphs that are not 𝑘−shifted−antimagic
Disconnected graphs
For any graph 𝐺, there exists a constant 𝑐=𝑐(𝐺) such that the graph 𝐺+𝑐 𝑃 3 is not antimagic.
The graph 𝑐 𝑃 3 is 𝑘-shifted-antimagic if and only if 𝑘∉ {− 5𝑐 2 ,− 5𝑐 ,…, 𝑐 2 −1}.
The graph 2 𝑃 4 is 𝑘-shifted-antimagic if and only if 𝑘∉{−2,−5}.
The graph 2 𝑆 3 is 𝑘-shifted-antimagic if and only if 𝑘∉{−2,−5}.

18. Open problems
Problem: Is it true that for any connected graph other than 𝐾 2 is 𝑘-shifted-antimagic, for sufficiently large integer |𝑘|?
Problem: Find a tree 𝑇 of diameter at least five which is not 𝑘-shifted-antimagic for some integer 𝑘.
Problem: Find a graph 𝐺 containing a cycle, which is not 𝑘-shifted-antimagic for some integer 𝑘.