Progress on Parallel Chip-Firing
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1 Progress on Parallel Chip-Firing Ziv Scully MIT PRIMES May 2, 2 Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
2 Motivation Simple rules Obvious patterns which are difficult to prove, or even wrong Potential connections to other fields of mathematics and science Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 2 / 8
3 Graphs Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 3 / 8
4 Graphs Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 3 / 8
5 Graphs Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 3 / 8
6 Graphs Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 3 / 8
7 Graphs Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 3 / 8
8 The Parallel Chip-Firing Game Played on a graph Assign a number of chips to each vertex On each turn: If a vertex has at least as many chips as neighbors, it fires Otherwise, we say it waits When a vertex fires, it gives one chip to each of its neighbors Happens for all vertices in parallel Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 4 / 8
9 The Parallel Chip-Firing Game c d b g a b c d e f g h 5 2 a 5 e f 2 h Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
10 The Parallel Chip-Firing Game c d b g a b c d e f g h 5 2 a 5 e f 2 h Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
11 The Parallel Chip-Firing Game c d b 2 g a b c d e f g h a 2 e 2 f h Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
12 The Parallel Chip-Firing Game c d b 2 g a b c d e f g h a 2 e 2 f h Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
13 The Parallel Chip-Firing Game c d b g a b c d e f g h a 3 e 2 f h Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
14 The Parallel Chip-Firing Game c d b g a b c d e f g h a 3 e 2 f h Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
15 The Parallel Chip-Firing Game c d 2 b g a b c d e f g h h a e 3 f Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
16 The Parallel Chip-Firing Game c d 2 b g a b c d e f g h h a e 3 f Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
17 The Parallel Chip-Firing Game c 2 d b g a b c d e f g h h a 2 e f 2 Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
18 The Parallel Chip-Firing Game c 2 d b g a b c d e f g h h a 2 e f 2 Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
19 The Parallel Chip-Firing Game c d b 2 g a b c d e f g h h a 2 e 2 f Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
20 The Parallel Chip-Firing Game c d b 2 g a b c d e f g h h a 2 e 2 f Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
21 Basic Properties All games are eventually periodic All vertices fire the same number of times in a period In a periodic- position, either all vertices fire or all vertices wait Period > 2 needs a cycle Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
22 Notation σ(t) is the position after taking t turns, starting with position σ() σ v (t) is the number of chips on vertex v in position σ(t) Φ v (t) is the number of v s neighbors that fire at time t; v gets one chip from each F v (t) is if v fires at time t and otherwise c is the total number of chips in a position If G is a graph, V (G) is its vertex set and E(G) is its edge set Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 7 / 8
23 Outline of Literature Bitar s conjecture: maximum period number of vertices Bitar and Goles: Trees have period or 2 Kiwi et al.: Bitar s conjecture is false! Dall Asta: Period on C n divides n Levine: Period on K n n Jiang: Period on K a,b 2 min(a, b) Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 8 / 8
24 Periodic or Not? c c d b g d b 2 g a 5 f 2 a 2 f e e 2 h Period 4 (not periodic) h Periodic-4 Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 9 / 8
25 Periodic-2 Positions Theorem (Characterization of periodic-2 positions) A position σ(t) on graph G is periodic-2 if and only if for all v V (G), deg(v) σ v (t) + Φ v (t) 2 deg(v). Proof. When the period is 2, vertices alternate between firing and waiting. The above inequality is true if and only if v is about to switch states. Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
26 Understanding Trees 7 8 Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
27 Understanding Trees 3 4 Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
28 Understanding Trees Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
29 Understanding Trees Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
30 Understanding Trees Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
31 Understanding Trees Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
32 Understanding Trees Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
33 Understanding Trees Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
34 Understanding Trees Period Period Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
35 Understanding Trees Period Period Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
36 Understanding Trees Period Period Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 / 8
37 Understanding Trees Theorem (Number of chips on a tree determines period) If a game on a tree graph G has c chips, its eventual period is 2 if and only if E(G) c 2 E(G). Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 2 / 8
38 Understanding Trees Theorem (Number of chips on a tree determines period) If a game on a tree graph G has c chips, its eventual period is 2 if and only if E(G) c 2 E(G). Proof. If the period is n, then for some time t, σ(t) will be periodic-n. If n = : σ v (t) deg(v) deg(v) σ v (t) c E(G) 2 E(G) c If n = 2: deg(v) σ v (t) + Φ v (t) 2 deg(v) 2 E(G) c + Φ v (t) + Φ v (t + ) 3 E(G) 2 E(G) c 2 E(G) Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 2 / 8
39 Firing Patterns String of s and s indicating firing and waiting, respectively Classification Alternating: (, ) Sparse: not alternating, two types Sparsely firing: never fires twice in a row Sparsely waiting: never waits twice in a row Clumpy: neither sparse nor alternating Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 3 / 8
40 Motors A special vertex with a fixed firing pattern Doesn t care about receiving chips Natural motors Subgraphs that follow normal chip firing rules One key vertex behaves like a motor Receiving external chips doesn t change its firing pattern Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 4 / 8
41 Motorized Trees some graph Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
42 Motorized Trees some graph motor Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
43 Motorized Trees some graph motor Theorem (Periodic behavior of trees with one sparse motor) If motor m in tree graph G is sparse, then for all v V (G) at any periodic time t, F v (t) = F m (t d), where d is the distance from m to v. Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
44 Motorized Trees some graph motor Theorem (Periodic behavior of trees with one sparse motor) If motor m in tree graph G is sparse, then for all v V (G) at any periodic time t, F v (t) = F m (t d), where d is the distance from m to v. Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
45 Motorized Trees some graph motor Theorem (Periodic behavior of trees with one sparse motor) If motor m in tree graph G is sparse, then for all v V (G) at any periodic time t, F v (t) = F m (t d), where d is the distance from m to v. Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
46 Motorized Trees some graph motor Theorem (Periodic behavior of trees with one sparse motor) If motor m in tree graph G is sparse, then for all v V (G) at any periodic time t, F v (t) = F m (t d), where d is the distance from m to v. Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
47 Motorized Trees some graph motor Theorem (Periodic behavior of trees with one sparse motor) If motor m in tree graph G is sparse, then for all v V (G) at any periodic time t, F v (t) = F m (t d), where d is the distance from m to v. Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
48 Motorized Trees some graph motor Theorem (Periodic behavior of trees with one sparse motor) If motor m in tree graph G is sparse, then for all v V (G) at any periodic time t, F v (t) = F m (t d), where d is the distance from m to v. Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
49 Motorized Trees some graph motor Theorem (Periodic behavior of trees with one sparse motor) If motor m in tree graph G is sparse, then for all v V (G) at any periodic time t, F v (t) = F m (t d), where d is the distance from m to v. Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 5 / 8
50 Constructing Natural Sparse and Alternating Motors 2 (each) 5 Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
51 Constructing Natural Sparse and Alternating Motors 2 5 (, Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
52 Constructing Natural Sparse and Alternating Motors 2 5 (,, Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
53 Constructing Natural Sparse and Alternating Motors 2 5 (,,, Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
54 Constructing Natural Sparse and Alternating Motors 2 5 (,,,, Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
55 Constructing Natural Sparse and Alternating Motors (,,,,, Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
56 Constructing Natural Sparse and Alternating Motors 2 (,,,,,, Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
57 Constructing Natural Sparse and Alternating Motors 2 5 (,,,,,,, Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
58 Constructing Natural Sparse and Alternating Motors 2 5 (,,,,,,, Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
59 Constructing Natural Sparse and Alternating Motors 2 5 (,,,,,,, ) Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 6 / 8
60 Further Questions Can a vertex have a clumpy firing pattern in a period? Can every vertex firing be traced back to a driving cycle? If a graph has a possible period of length mp for some prime p, must the graph have a cycle of length np? Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 7 / 8
61 Acknowledgments Mentor, Yan Zhang, MIT Collaborator, Damien Jiang, MIT MIT Program for Research in Mathematics, Engineering and Science Dr. Anne Fey, TU Delft Dr. Tanya Khovanova, MIT Dr. Lionel Levine, MIT Ziv Scully (MIT PRIMES) Progress on Parallel Chip-Firing May 2, 2 8 / 8
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