Antimagic Labellings of Graphs Torsten Mtze Joint work
Antimagic Labellings of Graphs Torsten Mütze Joint work with Dan Hefetz and Justus Schwartz
Outline Undirected Graphs n Directed Graphs n Selected Proof Ideas n
Motivation and Definition Magic Square 8 1 6 3 5 7 4 9 2 15 Antimagic Square 15 9 8 7 2 1 6 3 4 5 15 15 15 11 24 9 12 Antimagic graph labelling [Hartsfield, Ringel, 1990] - Undirected graph G=(V, E), m: =|E| - Labelling of the edges with 1, 2, …, m - All vertex sums distinct 15 14 13 18 15 7 6 20 3 6 8 2 21 4 1 5 10
Some Easy Observations [Hartsfield, Ringel, 1990] n n n 1 1 13 2 53 7 44 P 2 is not antimagic. Pn, Cn (n≥ 3) are antimagic. 1 13 2 53 8 594 4 Graphs of maximum degree n-1 are antimagic for n≥ 3 (e. g. Kn, Sn, Wn). u 2. Use remaining largest labels to achieve antimagic property 1. Use labels 1, 2, …, m-(n-1) arbitrarily v 1 vn-1 v 2 G-u … 1. w(v 1) ≤ w(v 2) ≤ … ≤ w(vn-1) 2. w(v 1) < w(v 2) < … < w(vn-1) < w(u)
More Results n Theorem [Hefetz, 2005]: A graph on 3 k vertices that admits a K 3 -factor is antimagic. n Theorem [Cranston, 2007]: Every regular bipartite graph with minimum degree 2 is antimagic. n Theorem [Alon, Kaplan, Lev, Roditty, Yuster, 2004]: Every graph with minimum degree W(log n) is antimagic. n Many other special cases are known n Conjecture [Hartsfield, Ringel, 1990]: Every connected graph but P 2 is antimagic.
What about Directed Graphs? -1 Antimagic labelling of directed graphs - Directed graph D=(V, E), m: =|E| - Labelling of the edges with 1, 2, …, m - All oriented vertex sums distinct 7 6 -14 3 8 2 11 4 5 1 8 n Question 1: Given an undirected graph G, is every orientation D(G) antimagic? n Question 2: Given an undirected graph G, is there an antimagic orientation D(G)? ∀ ∃ -4
First Observations n P 2 has an antimagic orientation. n There are directed graphs that are not antimagic. -1 -1 2 1 -1 1 2 2 2 3 -1 n Conjecture: Every connected directed graph with at least 4 vertices is antimagic. n Lemma: If G is an undirected bipartite antimagic graph, then G has an antimagic orientation. w(a 1) w(a 2) w(a 3) w(a 4) - w(b 1) - w(b 2) - w(b 3)
Our Results n Theorem 1: Every orientation of every undirected graph with minimum degree W(log n) is antimagic. n Theorem 2: Every orientation of every undirected graph in one of the following families is antimagic: stars Sn, wheels Wn, cliques Kn (n≥ 4). n Theorem 3: Let G be a (2 d+1)-regular undirected graph. Then there exists an antimagic orientation of G. (extension to 2 d-regular graphs is possible with slight extra conditions) ∀ ∃
Proof Strategies Explicit labellings - (Almost) magic labelling + distortion - Subset control n Algebraic techniques - Combinatorial Nullstellensatz n Probabilistic methods - Lovász Local Lemma n
Proof of Theorem 3 n Theorem 3: Let G be a (2 d+1)-regular undirected graph. Then there exists an antimagic orientation of G. (extension to 2 d-regular graphs is possible with slight extra conditions) n Proof idea: (Almost) magic labelling + distortion -1 12 -1 13 15 G 7 6 4 10 -1 -2 -2 -1 9 5 14 2 -1 8 10 1 13 0 3 M 11 15 -2 1 14 -1 -1 -1 -2 -2 5 4 3 4 9 -2 -12 -7 -6 -3 1. Add perfect matching M to connect different components 2. Orient and label along some eulerian cycle (use duplicate labels for M-edges) 3. Remove matching edges and obtain an antimagic labelling 13 -2
Proof of Theorem 2 (cliques) n n Theorem 2: Every orientation of Kn (n≥ 4) is antimagic. Proof idea: Subset control + parity argument u G-u H vn-1 v 2 m=(2) n 1 w(v 1) < … < w(vr) w(v 1) ≤ … ≤ w(vr) … … r vr vr+1 n-1 -r w(vr+1) ≤ … ≤ w(vn-1) w(vr+1) < … < w(vn-1) 1. Pick vertex u of maximum in-degree (r: =deg+(u), deg-(u)=n-1 -r) 2. Reserve the r largest labels of the same parity and the n-1 -r smallest labels of the opposite parity 3. Distribute remaining odd labels over even degree subgraph H of G-u 4. Distribute remaining even labels over remaining edges of G-u 5. Achieve antimagic property in each class and by parity among all vertices
Proof by Probabilistic Methods (1) n Theorem [Alon, Kaplan, Lev, Roditty, Yuster, 2004]: Every graph with minimum degree W(log n) is antimagic. Proof can be adapted for every orientation of every undirected graph with minimum degree W(log n) (Theorem 1) n Proof idea: Lovász Local Lemma A 1, A 2, …, As events Pr[Ai] ≤ p Ai independent of all but at most r other events p(r+1) < ⅓ A(u, v) : = vertex sums of u and v are the same Naive: random permutation of the edge labels All events are dependent, r = (n 2)-1 ≈ n 2 With positive probability no event Ai holds With positive probability no two vertex sums are the same Antimagic labelling
Proof by Probabilistic Methods (2) Assume G is d-regular (d=C log n) and has an even number of edges m 1 {10, 17} v u u 2 d possible choices 2 random phases 1. Partition edges into pairs, such that no pair shares an endvertex {3, 5} {11, 13} S(u) = {24, 26, 28, 31, 33, 35} Pr[w(u)=k] = ⅛ ¼ ⅛ 2. Randomly partition label set into pairs and assign to edge pairs 3. Fix a partition from 2, such that at all vertices every value from S(u) is obtained only by a small fraction among the 2 d choices: Pr[w(u)=k] ≤ small 4. Flip coin for each edge pair which edge gets which label Pr[A(u, v)] = Pr[w(u)=k] ≤ small 5. Dependencies: r = (6 d+2)n ≈ 6 dn n 2 6. LLL: p(r+1) = small (6 dn+1) ⅓ u v
Questions?
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