Strange Combinatorial Connections Tom Trotter Georgia Institute of Technology trotter@math.gatech.edu February 13, 2003
Proper Graph Colorings Definition. A proper r- coloring of a graph G is a map φ from the vertex set of G to {1, 2,..., r} so that φ(x) φ(y) whenever x and y are adjacent vertices in G. 1 2 5 2 4 2 4 8 2 1 3 6 1 3 2 2 1 1 3 2
Chromatic Number Definition. The chromatic number χ(g) is the fewest number of colors required for a proper coloring of G. 1 2 5 2 4 2 4 5 2 1 3 4 1 3 5 2 1 1 5 2
Partially Ordered Sets (Posets) Definition. A partially ordered set P is a pair (X, P ) where X is a set and P is a reflexive, antisymmetric and transitive binary relation on X.
Interval Orders Definition. A poset P = (X, P ) is an interval order if for every x X, there is a closed interval [a x, b x ] of the real line R so that x < y in P if and only if b x < a y in R. d c b y b a x y c d x a
Sets and Inclusion abdefh fg abcdf abeh f abde abcd ac abe bd a b
Subset Lattices - Cubes abcd abc abd acd bcd ab ac bc ad bd cd a b c d
Chromatic Number of Poset Diagrams Proposition. If P is a poset of height h, then the chromatic number of the diagram of P is at most h.
A Ramsey Theoretic Construction Theorem. [Nešetrǐl and Rődl] For every h 1, there exists a poset P of height h so that the chromatic number of the diagram of P is h.
A Ramsey Theoretic Construction Theorem. [Nešetrǐl and Rődl] For every h 1, there exists a poset P of height h so that the chromatic number of the diagram of P is h. The preceding theorem is an immediate consequence of a more general result which provides an explicit construction of simple hypergraphs with large girth and large chromatic number.
Strange Problem Number 1 Problem. For each t 1, find the largest integer h = C(t) so that whenever P is an interval order of height h, the chromatic number of the diagram of P is at most t.
Strange Problem Number 1 Problem. For each t 1, find the largest integer h = C(t) so that whenever P is an interval order of height h, the chromatic number of the diagram of P is at most t. Remark. C(t) > c t. It is not too difficult to show that there exists a constant c > 1 so that
Strange Problem Number 2 Definition. A sequence (S 0, S 1,..., S h ) of sets is called an F -sequence if 1. S 0 S 1, and 2. S j S i S i+1 when j > i + 1.
Strange Problem Number 2 Definition. A sequence (S 0, S 1,..., S h ) of sets is called an F -sequence if 1. S 0 S 1, and 2. S j S i S i+1 when j > i + 1. Problem. For each positive integer t, find the largest positive integer h = F (t) for which there exists an F -sequence (S 0, S 1,..., S h ) of subsets of {1, 2,..., t}.
Examples Example. (, {1}, {2}, {3}, {1, 3}, {1, 2, 3}) is an F sequence of subsets of {1, 2, 3}.
Examples Example. (, {1}, {2}, {3}, {1, 3}, {1, 2, 3}) is an F sequence of subsets of {1, 2, 3}. So F (3) 5.
Examples Example. (, {1}, {2}, {3}, {1, 3}, {1, 2, 3}) is an F sequence of subsets of {1, 2, 3}. So F (3) 5. In fact, F (3) = 5.
Examples Example. (, {1}, {2}, {3}, {1, 3}, {1, 2, 3}) is an F sequence of subsets of {1, 2, 3}. So F (3) 5. In fact, F (3) = 5. Example. (, {1}, {2}, {3}, {4}, {2, 4}, {1, 4}, {1, 3}, {1, 2, 3}, {2, 3, 4}) is an F sequence of subsets of {1, 2, 3, 4}.
Examples Example. (, {1}, {2}, {3}, {1, 3}, {1, 2, 3}) is an F sequence of subsets of {1, 2, 3}. So F (3) 5. In fact, F (3) = 5. Example. (, {1}, {2}, {3}, {4}, {2, 4}, {1, 4}, {1, 3}, {1, 2, 3}, {2, 3, 4}) is an F sequence of subsets of {1, 2, 3, 4}. So F (4) 9.
Examples Example. (, {1}, {2}, {3}, {1, 3}, {1, 2, 3}) is an F sequence of subsets of {1, 2, 3}. So F (3) 5. In fact, F (3) = 5. Example. (, {1}, {2}, {3}, {4}, {2, 4}, {1, 4}, {1, 3}, {1, 2, 3}, {2, 3, 4}) is an F sequence of subsets of {1, 2, 3, 4}. So F (4) 9. In fact, F (4) = 9.
Long F -Sequences Lemma. [Felsner and Trotter] If k and t are positive integers with k < t and h = 1 + ( ) t 1 k, then there exists an F -sequence (S0, S 1,..., S h ) of k-element subsets of {1, 2,..., t} so that S i S i+1 = 2 for all i = 1, 2,..., h 1
Long F -Sequences Lemma. [Felsner and Trotter] If k and t are positive integers with k < t and h = 1 + ( ) t 1 k, then there exists an F -sequence (S0, S 1,..., S h ) of k-element subsets of {1, 2,..., t} so that S i S i+1 = 2 for all i = 1, 2,..., h 1 Corollary. [Felsner and Trotter] For all t 1, F (t) > 2 t 2.
Hamiltonian Cycles in Cubes Proposition. For t 2, the t-cube has a hamiltonian cycle.
Hamiltonian Cycles in Cubes Proposition. For t 2, the t-cube has a hamiltonian cycle. abcd abc abd acd bcd ab ac bc ad bd cd a b c d
Corresponding Edges Then select any pair of corresponding edges: abcd abc abd acd bcd ab ac bc ad bd cd a b c d
Exchange edges as shown: A hamiltonian cycle in the t + 1-cube abcd abc abd acd bcd ab ac bc ad bd cd a b c d
Monotone Hamiltonian Paths Definition. Let t be a positive integer and let n = 2 t. A listing (A 1, A 2,..., A t ) of the subsets of {1, 2,..., t} is called a monotone hamiltonian path in the t-cube if 1. A 1 =, and 2. If 1 < i < t and S A i, then S = A j for some j {1, 2,..., i + 1}.
The 2-Cube - Getting Started ab a b
The 2-Cube - Must Go Up ab a b
The 2-Cube - Last Move is Forced ab a b
The 3-Cube - Getting Started abc ab ac bc a b c
The 3-Cube - Must Go Up abc ab ac bc a b c
The 3-Cube - Must Go Back Down abc ab ac bc a b c
The 3-Cube - Forced Move abc ab ac bc a b c
The 3-Cube - SUCCESS!! abc ab ac bc a b c
The 4-Cube - Getting Started abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Try Left abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Right is Illegal!! abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Must go Up abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Only Move is Illegal!! abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Backtrack to Here abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Try Right abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Try Left abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Straight Up Illegal!! abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Right is Illegal Too! abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Backtrack to Here abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Must go Straight Up abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Right Doesn t Work abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Backtrack to Here abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Try Left abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - Must go Right abcd abc abd acd bcd ab ac bc ad bd cd a b c d
The 4-Cube - SUCCESS!! abcd abc abd acd bcd ab ac bc ad bd cd a b c d
A Monotone Hamiltonian Path in the 5-Cube
Strange Problem Number 3 Conjecture. t-cube. For every t 1, there exists a monotone hamiltonian path in the
Computational Results The following table shows the number H(t) of monotone hamiltonian paths in the t-cube: H(1) = 1 H(2) = 1 H(3) = 1 H(4) = 1 H(5) = 10 H(6) = 123 H(7) = 1492723
More Computational Results Also, H(8) > 10 14
More Computational Results Also, H(8) > 10 14 H(9) > 0
More Computational Results Also, H(8) > 10 14 H(9) > 0 H(10) > 0
Combinatorial Connections Recall that 1. C(t) is the largest integer h so that whenever P is an interval order of height h, the chromatic number of the diagram of P is at most t.
Combinatorial Connections Recall that 1. C(t) is the largest integer h so that whenever P is an interval order of height h, the chromatic number of the diagram of P is at most t. 2. F (t) is the largest h for which there exists an F -sequence (S 0, S 1,..., S h ) of subsets of {1, 2,..., t}.
Combinatorial Connections Recall that 1. C(t) is the largest integer h so that whenever P is an interval order of height h, the chromatic number of the diagram of P is at most t. 2. F (t) is the largest h for which there exists an F -sequence (S 0, S 1,..., S h ) of subsets of {1, 2,..., t}. Theorem. [Felsner and Trotter] For every t 1, C(t) = F (t).
A Natural Partition into Antichains Let P = (X, P ) be a poset of height h. Then for each x X, let d(x) denote the largest positive integer i for which there exists a chain x = x 1 < x 2 < < x i in P.
A Natural Partition into Antichains Let P = (X, P ) be a poset of height h. Then for each x X, let d(x) denote the largest positive integer i for which there exists a chain x = x 1 < x 2 < < x i in P. Remark. For each i = 1, 2,..., h, the set X i = {x X : d(x) = h + 1 i} is an antichain in P and is a partition into antichains. X = X 1 X 2 X h
Special Property of Interval Orders Remark. Let x be a point in an interval order P. Then any two points covered by x differ in depth by at most 1. x
Using F -sequences to Color Diagrams Here s the proof that C(t) F (t) for all t 1.
Using F -sequences to Color Diagrams Here s the proof that C(t) F (t) for all t 1. 1. Vertices from X j will get colors from S j.
Using F -sequences to Color Diagrams Here s the proof that C(t) F (t) for all t 1. 1. Vertices from X j will get colors from S j. 2. For each x X j, choose a color from S j which does not belong to S i S i+1 where all points covered by x come from X i X i+1.
Using F -sequences to Color Diagrams Here s the proof that C(t) F (t) for all t 1. 1. Vertices from X j will get colors from S j. 2. For each x X j, choose a color from S j which does not belong to S i S i+1 where all points covered by x come from X i X i+1. Showing that F (t) C(t) for all t 1 is somewhat more complicated and requires a Ramsey theoretic argument.
F -Sequences and Monotone Hamiltonian Paths Lemma. [Felsner and Trotter] If there is a monotone hamiltonian path in a t-cube, then there is an F -sequence (S 0, S 1,..., S h ) of subsets of {1, 2,..., t} with h = 2 t 1 + t 1 2.
Example abcd abc abd acd bcd ab ac bc ad bd cd a b c d
Example abcd abc abd acd bcd ab ac bc ad bd cd a b c d Example. (, {a}, {b}, {c}, {d}, {b, d}, {a, d}, {a, c}, {a, b, c}, {b, c, d}) is an F sequence of subsets of {a, b, c, d}.
F -Sequences and Monotone Hamiltonian Paths Theorem. [Felsner and Trotter] For each t 1, F (t) 2 t 1 + t 1 2 with equality holding if and only if there is a monotone hamiltonian path in the t-cube.
Partial Results Remark. If there is a monotone hamiltonian path in the t-cube, then for odd k, there must exist an F -sequence (S 0, S 1,..., S h ) of k-element subsets of {1, 2,..., t} so that t = 1 + ( ) t 1 k and Si S i+1 = 2 for all i = 1, 2,..., h 1
Partial Results Remark. If there is a monotone hamiltonian path in the t-cube, then for odd k, there must exist an F -sequence (S 0, S 1,..., S h ) of k-element subsets of {1, 2,..., t} so that t = 1 + ( ) t 1 k and Si S i+1 = 2 for all i = 1, 2,..., h 1 Remark. Such F -sequences exist for all t and all k, both even and odd.
1. Shift Graphs Other Combinatorial Connections
Other Combinatorial Connections 1. Shift Graphs 2. Dedekind s problem
Other Combinatorial Connections 1. Shift Graphs 2. Dedekind s problem 3. Dimension of graphs
Other Combinatorial Connections 1. Shift Graphs 2. Dedekind s problem 3. Dimension of graphs 4. Correlation Inequalities
The Evidence is Overwhelming Recall that: H(1) = 1 H(2) = 2 H(3) = 6 H(4) = 24 H(5) = 1440 H(6) = 88560 H(7) = 7523323920
Nevertheless...
Nevertheless... Conjecture. [WTT s Wild One] If t is sufficiently large, H(t) = 0, there are no monotone hamiltonian paths in the t-cube.