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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
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Magic 1 2 3 4 5 6 7 8 9 Magic square 河圖洛書
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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
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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
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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.
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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.
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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.
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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.
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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
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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.
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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
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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
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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. 101 102 103 104 105 412 416 108 110 109 111 112 113 114 221 333 115 116 454 227 117 118 119 120 121 352 122 123 124 125 244 242 481 126 127 128 129 130 131 254 501 132 133 134 135 265 258 519 263 538 101 108 102 103 106 323 107 104 109 105
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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: ≤ ≤ ≤ ≤
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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.
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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}.
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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}.
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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.
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