Anti-magic labeling of graphs

Yu-Chang Liang

April 28, 2017

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Outline

1

Introduction

2

Tool

3

Known results

4

Our research

5

Further work

6

References

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History Definition Magic square is an arrangement of numbers in a square grid, where the numbers in each row, and in each column, and the numbers in the forward and backward main diagonals, all add up to the same number.

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Labeling

A graph labeling is a mapping f satisfied: f : A −→ B. S where A may be V (G), E(G), F (G), V (G) E(G), etc, and B be the set of numbers( the element of a group). Goal: The structure of φf (x) which is the function depend with f . P e.g., φf (v) = e∈E(v) f (e), or φf (uv) = f (u) − f (v)

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History Definition (Sedl´aˇcek 1963) A graph G is called magic if G has a labeling of the edges with distinct integers such that for each vertex v the sum of the labels of all edges incident with v is the same for all v.

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History

Definition (Stewart 1966, 1967) A graph G is called super-magic if G has a labeling of the edges with distinct consecutive integers such that for each vertex v the sum of the labels of all edges incident with v is the same for all v.

Definition (Stewart 1966, 1967) A graph G is called semi-magic if G has a labeling of the edges with integers such that for each vertex v the sum of the labels of all edges incident with v is the same for all v.

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History

Definition An S-weighting of a graph G is a map w : E(G) −→ S. k-weighting is an S-weighting with S = {1, . . . , k}.

Definition

P Let φw (v) = e∈E(v) w(e). A weighting w is proper if φw (v) is a proper coloring of G.

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History

Conjecture ((1,2,3)-Conjecture, 2004) Every graph without isolated edges has a proper 3-weighting.

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History

Definition An total S-weighting of a graph G is a map w : E(G) k-weighting is an S-weighting with S = {1, . . . , k}.

S

V (G) −→ S.

Definition

P Let φw (v) = w(v) + e∈E(v) w(e). A weighting w is proper if φw (v) is a proper coloring of G.

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History

Conjecture (1-2-Conjecture, 2007) Every graph has a proper total 2-weighting.

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History Definition (Rosa, A 1967) A graph G is called graceful if f is an injection from the vertices of G to the set {0, 1, . . . E(G)} such that, when each edge xy is assigned the label | f (x) − f (y) |, the resulting edge labels are distinct.

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History

Conjecture (1967) All trees are graceful.

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Definition Definition (Hartsfild and Ringel 1990) An anti-magic labeling of G is a bijection from the set of edges to the set {1, 2, . . . , |E(G)|}, such that all the vertex-sums are pairwise distinct, where the vertex-sum s(v) of a vertex v is the sum of labels of all edges incident with v.

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Tool

1 : Algorithm. (Strategy) 2 : Combinatorial Nullstellensatz. 3 : Probability method.

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Strategy

1 : Consider the structure of G. 2 : Design the function φf (v). 3 : characterize the graph G and f .

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Strategy of paths

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Strategy of paths

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Strategy of cycles

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Strategy of cycles

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Combinatorial Nullstellensatz

Let f be the labeling of G, and f (ei ) = xi , then g(x1 , x2 , . . . , xm ) =

Y

(xi − xj )

1≤i
Y

(φf (vi ) − φf (vj )).

1≤i
If there are x1 , x2 , . . . , xm , such that g(x1 , x2 , . . . , xm ) 6= 0, then G is anti-magic.

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Combinatorial Nullstellensatz

Let η be the index function. g(x1 , x2 , . . . , xm ) =

X η

cη

Y

η(i)

xi

.

i

If cη 6= 0, and |Si | ≥ η(i) + 1, then there are xi ∈ Si , such that g(x1 , x2 , . . . , xm ) 6= 0

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Combinatorial Nullstellensatz

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Combinatorial Nullstellensatz

g(x1 , x2 , . . . , x5 ) = (x1 + x2 − x2 − x3 )(x2 + x3 − x3 − x4 ) (x3 + x4 − x4 − x5 )(x4 + x5 − x5 − x1 )(x5 + x1 − x1 − x2 ) (x1 + x2 − x3 − x4 )(x2 + x3 − x4 − x5 )(x3 + x4 − x5 − x1 ) Y (x4 + x5 − x1 − x2 )(x5 + x1 − x2 − x3 ) (xi − xj ). 1≤i
The coefficient of x41 x42 x43 x44 x45 is −980

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Conjecture

Conjecture (Hartsfild and Ringel 1990) All trees except K2 have an anti-magic labeling.

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Conjecture

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Conjecture

Conjecture (Hartsfild and Ringel 1990) All connected graphs except K2 have an anti-magic labeling.

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Conjecture

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Trees

Theorem (Chawathe and Krishna 2002) Every complete m-ary tree is anti-magic.

Theorem (Kaplan, Lev and Roditty 2009) Every tree with at most one vertex of degree 2 is anti-magic. (The proof has an error)

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Regular graphs

Theorem (Hartsfild and Ringel 1990) Every 2-regular graph is anti-magic.

Theorem (Cranston 2009) Every k-regular bipartite graph is anti-magic.

Theorem (T-M Wang and G-H Zhang 2013) Regular Graph with Particular Factors is anti-magic.

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Cartesian product graphs

Theorem (T-M Wang 2005) Let G be the Cartesian product of cycles Cn1 Cn2  · · · Cnk , then G is anti-magic.

Theorem (T-M Wang 2005) If R is an r-regular anti-magic graph with r > 1 then RCn is anti-magic.

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(ω, k)-antimagic

Theorem (Wong and Zhu 2011) If G has a universal vertex, then G is weighted-2-antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted-1-antimagic.

Theorem (Wong and Zhu 2011) If G has a prime number of vertices and has a Hamiltonian path, then G is weighted-1-antimagic.

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(ω, k)-antimagic

Theorem (Hefetz 2005) Let G be a graph with q edges and at most one isolated vertex and no isolated edges is(ω, 2q − 4)-antimagic.

Theorem (Hefetz 2005) Let G be a graph with p > 2 vertices that admits a 1-factor is (p − 2)-antimagic.

Theorem (Hefetz 2005) Let G be a graph with p vertices and maximum degree n − k, where k ≥ 3 is any function of p is (3k − 7)-antimagic

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Cartesian product graphs

Theorem (Zhang and Sun 2009) If R is an anti-magic regular graph then for any graph G, GR is anti-magic.

Theorem (Zhang and Sun 2009) Let G be the Cartesian product of many paths Pn1 Pn2  · · · Pnt , then G is anti-magic, provided that at least one of the paths has length at last 3.

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Tree

Theorem (Chawathe and Krishna 2002) Every complete m-ary tree is anti-magic.

Theorem (Kaplan, Lev and Roditty 2009) Every tree with at most one vertex of degree 2 is anti-magic. (The proof has an error)

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General graphs Theorem (Yilma 2013) If G is a graph with maximum degree ∆ = |V (G)| − 3, and |V (G)| ≥ 9 then G is anti-magic.

Theorem (Hefetz, Saluz and Tran 2005, 2010) If G is a graph with pk vertices which admits a Cp -factor, then G is anti-magic.

Theorem (Alon, Kaplan, Lev, Roditty and Yuster 2004) If G is a graph with minimum degree δ(G) ≥ C log |V (G)|, then G is anti-magic.

Theorem (Eccles 2014) The graphs of large linear size are anti-magic. Liang ()

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Survey

[Joseph A. Gallian 1997 2016] A Dynamic Survey of Graph Labeling.

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Results

Theorem (Liang, Wong and Zhu 2014) If T 6= K2 is a tree with at most one vertex of degree 2, then T is anti-magic.

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Consider the structure of tree T .

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Design the function φf (v).

Each vertex has at most one in-edge. Goal: the labeling of in-edge are distinct implies φf (v) are distinct.

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characterize the graph T and f . The sum of labeling of the out-edge are the same(based on the group).

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Trees Definition Suppose G is a graph and e = (u, v) is an edge of G. A subdivision of e is the operation of replacing e = (u, v) with a path (u, we , v), where we is a new vertex.

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Trees

Definition Suppose T = (V, E) is a tree and A ⊆ E. Denote by T ∗ (A) the tree obtained by subdividing each edge in A. Let T ∗ = T ∗ (E) be the tree which obtained by subdivide each edge of T .

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Trees Theorem (Liang, Wong and Zhu 2014) If T is a tree with V2 (T ) = ∅ and T ∗ is obtained from T by subdividing every edge, then T ∗ is anti-magic.

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Trees Theorem (Liang, Wong and Zhu 2014) If T is a tree in which every vertex v ∈ V (T ) has an even number of sons, then for any A ⊆ E(T ), T ∗ (A) is obtained from T by subdividing every edge in A is anti-magic.

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Trees Definition Let Vi (T ) beSthe set of vertices of T of degree i, and Veven (T ) = i>0 V2i (T ). If Veven (T ) induces a path, then we assume Veven (T ) = (v1 , v2 , . . . , vk−1 ), and P (v0 , v1 , . . . , vk ) is also a path.

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Trees Theorem (Liang, Wong and Zhu 2014) Assume T is a tree, V2 (T ) induces a path Pk−1 and each component of T \ V2 (T ) is arbitrary complete d-ary tree but not a star, then T is anti-magic.

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Trees Theorem (Liang, Wong and Zhu 2014) Assume T is a tree, Veven (T ) induces a path and |Veven (T )| is odd, then T is anti-magic.

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Trees Theorem (Liang, Wong and Zhu 2014) Assume T is a tree, Veven (T ) induced a path, and |Veven (T )| is even. Let 2s = |Veven (T )|. If deg(vs ) 6= deg(vk ) + 1 or deg(vs+1 ) 6= deg(v0 ) + 1, then T is anti-magic.

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Cartesian product graphs

Theorem (Liang and Zhu 2013) Assume G is a k-regular graph with k ≥ 2. For any connected graph H with |E(H)| ≥ |V (H)| − 1 ≥ 1, HG is anti-magic.

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Cartesian product graphs

Theorem (Liang and Zhu 2013) Assume H is a graph with |E(H)| ≥ |V (H)| − 1. If each component of H has a vertex of odd degree, then the prism of H is anti-magic.

Theorem (Liang and Zhu 2013) If H is a graph with m ≥ 2n − 2, where m = |E(G)|, and n = |V (G)|, then the prism of H is anti-magic.

Theorem (Liang and Zhu 2013) If G is a graph with ∆(G) = |V (G)| − 1, GP2 is anti-magic.

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Cartesian product graphs Theorem (Liang and Zhu 2013) Assume G is obtained from a regular graph by adding a universal vertex u, then for any positive integer k, Gk is anti-magic.

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Cartesian product graphs

Theorem (Liang and Zhu 2013) If each of G1 and G2 is obtained from a (d − 1)-regular graph by adding a universal vertex, then G1 G2 is anti-magic.

Theorem (Liang and Zhu 2013) If G1 , G2 are double star graphs, then G1 G2 is anti-magic.

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Regular graphs

Theorem (Liang and Zhu 2014, Cranston, Liang and Zhu 2015, Chang, Liang, Pan and Zhu 2015 ) Let k > 1 be a positive integer, and G is a k-regular graph, then G is anti-magic.

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idea

Definition : Given v ∗ is a vertex of G, let Li = {u : dG (u, v ∗ ) = i}. Goal 1 : For any u, v ∈ Li , s(v) 6= s(u). Goal 2 : For any x ∈ Li−1 , y ∈ Li , s(x) > s(y). Idea : For each vertex v ∈ Li , N (v) ⊆ Li−1

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S

Li

S

Li+1 .

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Partition

Partition : Fix the vertex v ∗ of G, Let L0 S = {v ∗ } and for i ≥ 1, ∗ Li = {u : dG (u, v ) = i}. Assume V (G) = pi=0 Li .

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Partition

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Labeling Let Ei = E(G[Li ]) ∪ E(G[Li−1 , Li ]). We label the edges in Ep , Ep−1 , . . . , E1 recursively in this order.

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Labeling The partial-sum ω(v) of a vertex v is the sum of labels of all edges incident with v except σ(v). Ordering {ω(v)|v ∈ Lj } by {ωi |1 ≤ j ≤ |Lj |} such that ωi ≤ ωi+1 , for all i.

So for any u, v ∈ Li , s(v) 6= s(u). Liang ()

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Proof of theorem

We label the open trail in Gσ [Li , Li−1 ] by [s, l] as follow:

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Further work

1 : Tree. Cartesian product graph. Halin graph. Cartesian power graph. 2 : (ω, k)- anti-magic. 3 : New tool for general graph. 4 : Hyper graph version. 5 : Directed graph version.

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References N. Alon, G. Kaplan, A. Lev, Y. Roditty, and R. Yuster, Dense graphs are antimagic, J. Graph Theory, 47 (2004), 297-309. A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967) 349-355. Y. Cheng, A new class of antimaig Cartesian product graphs, Discrete Math., 308 (2008), 6441-6448. Y. Cheng, Lattice grids and prisms are antimagic, Theoret. Comput. Sci., 374 (2007), 66-73. P. D. Chawathe, and V. Krishna, Antimagic labelings of complete m-ary trees, Number theory and discrete mathematics (Chandigarh, 2000), 77-80, Trends Math., Birkh¨ auser, Basel, 2002. D. W. Cranston, Regular bipartite graphs are antimagic, J. Graph Theory, 60 (2009), 173-182. Liang ()

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References

D. W. Cranston, Y. Liang and X. Zhu, Regular Graphs of Odd Degree Are Antimagic, J Graph Theory, 80 (2015), 28-33. T. Eccles, Graphs of large linear size are antimagic, manuscript, 2014, arXiv:1409.3659 J. A. Gallian, A Dynamic survey on Graph Labeling, The Electronic Journal of Combinatorics, 15 (2008). S. W. Golomb, How to number a graph, Graph theory and computing, 23-37. Academic Press, New York, 1972. P. Hall , On representatives of subsets, J.Lond. Mat. Sc., 10 (1935), 26-30.

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References N. Hartsfield, and G. Ringel, Pearls in Graph Theory, Academic Press, INC., Boston, 1990, pp. 108-109, Revised version 1994. D. Hefetz, Anti-magic graphs via the Combinatorial Nullstellensatz, J. Graph Theory, 50 (2005), 263-272 D. Hefetz, H.T.T. Tran, and A. Saluz, An application of the Combinatorial Nullstellensatz to a graph labelling problem, J. Graph Theory, 65 (2010), 70-82. G. Kaplan, A. Lev, and Y. Roditty, On zero-sum partitions and anti-magic trees, Discrete Math., 309 (2009), 2010-2014. Y. Liang, T. Wong, and X. Zhu, Anti-magic labeling of trees, Discrete Math. 331 (2014), 9-14. Y. Liang, and X. Zhu, Anti-magic labeling of cubic graphs, J. Graph Theory, 75 (2014), 31-36. Liang ()

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References Y. Liang, and X. Zhu, Anti-magic labelling of Cartesian product of graphs, Theoret. Comput. Sci., 477 (2013), 1-5. G. Ringel, Problem 25, in Theory of Graphs and its Applications, Proc. Symposium Smolenice 1963, Prague (1964), 162. A. Rosa, On certain valuations of the vertices of a graph, 1967 Theory of Graphs (Internat. Sympos., Rome, 1966) pp. 349-355 Gordon and Breach, New York; Dunod, Paris. J. Sedl´aˇcek, Problem 27, in Theory of Graphs and its Applications, Proc. Symposium Smolenice, June, (1963) 163-167. T. M. Wang, Toroidal grids are antimagic, computing and combinatorics, Lecture Notes in Comput. Sci., 3595, Springer, Berlin(2005), 671-679. Liang ()

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References

T. M. Wang, and C. C. Hsiao, On antimagic labeling for graph products, Discrete Math., 308 (2008), 3624-3633. D. B. West, Introduction to Graph Theory, Second Edition, Prentice Hall, Inc., Upper Saddle River, NJ, 1996. T. Wong, and X. Zhu, Antimagic labelling of vertex weighted graphs , J. Graph Theory, 70 (2012), 348-359. Z. B. Yilma, Antimagic properties of graphs with large maximum degree , J. Graph Th., 72 (2013), no. 4, 367-373. Y. Zhang, and X. Sun, The antimagicness of the Cartesian product of graphs, Theoret. Comput. Sci., 410 (2009), 727-735.

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Thanks for your attention!

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