The Chromatic Symmetric Function: Hypergraphs and Beyond

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1 The : Hypergraphs and Beyond arxiv: Department of Mathematics Michigan State University Graduate Student Combinatorics Conference, April 07

2 Outline Introducing the chromatic symmetric function Definition of the chromatic symmetric function The power sum expansion Uniform hypertrees Recovering the degree sequence 3 Oriented coloring Acyclic coloring

3 Definition of the chromatic symmetric function The power sum expansion Defining of the chromatic symmetric function Let H = (V, E) be a hypergraph and let P = {,... } denote the positive integers. A map f : V P is a proper coloring of H if no hyperedge is monochromatic. Let x, x,... be commuting variables, then any proper coloring f determines a monomial x f := v V x f (v) and the chromatic symmetric function of H denoted X H is defined by X H := f x f where the sum runs over all proper colorings.

4 Definition of the chromatic symmetric function The power sum expansion An example chromatic symmetric function Consider the hypergraph H = ({,, 3}, {{,, 3}}) shown below. 3 An example of a proper coloring is f () =, f () =, and f (3) = with corresponding monomial x f = x x. Another proper coloring is g() =, g() =, and g(3) = corresponding to the monomial x g = x x. We have the following expansion of X H in the m-basis: X H = 6m,, + 3m,

5 Definition of the chromatic symmetric function The power sum expansion The power sum expansion Let H = (V, E) be a hypergraph. For any A E let λ(a) denote the integer partition given by the sizes of the connected components of (V, A). The chromatic symmetric function then expands as X H = A E( ) A p λ(a). Looking again at H = ({,, 3}, {{,, 3}}), we have λ( ) =,, and λ({{,, 3}}) = 3. 3 So, we can verify X H = p,, p 3 = 6m,, + 3m, in this case.

6 Uniform hypertrees Recovering the degree sequence Defining hypertrees A hypergraph H = (V, E) is a hypertree if it is connected and acyclic. In particular hypertrees must be linear (i.e. for all e e E, e e ) Figure: A hypertree 3 Figure: Not a hypertree (not linear) Figure: Not a hypertree

7 Uniform hypertrees Recovering the degree sequence Uniform hypertrees A hypergraph H = (V, E) is called s-uniform if e = s for all e E. Proposition (Borowiecki - Lazuka 007) If H = (V, E) is a linear hypergraph on n vertices, then H is an s-uniform hypertree if and only if P H = t(t s ) m where E = m and P H denotes the chromatic polynomial. Question Does the chromatic symmetric function distinguish s-uniform hypertrees? For s = the above question is a well studied open problem.

8 Uniform hypertrees Recovering the degree sequence Degree sequences of trees It is known from computer calculations of Russell (0) that the chromatic symmetric function is a complete invariant among trees of 5 or fewer vertices. We also have the following result which we will be able to extend to uniform hypertrees. Corollary (Martin-Morin-Wagner 008) If T is a tree, then the degree sequence of T can be obtained from X T. Remark The above result is a corollary of the stronger result that the chromatic symmetric function of a tree determines its subtree and connector polynomials.

9 Uniform hypertrees Recovering the degree sequence Degree sequences of hypertrees Theorem If T is an s-uniform hypertree, then the degree sequence of T can be obtained from X T. Proof. (Sketch) Use the power sum expansion of X T. Looking the of s in the partitions λ with appearing of p λ appearing in X T we obtain a triangular system. We solve this system to determine the number of vertices of each degree. Full proofs are hopeless in a 0 minute talk, let s try an example...

10 Uniform hypertrees Recovering the degree sequence Example hypertree degree sequence Let T be the hypertree above, then edge 3 edges {}}{{}}{ X T = p 7 3p }{{} 3, + p 5, + p 3, p 7 }{{} 0 edges edges Let D i = #(vetices of degree i) in T. D 3 + D = + = 5 ( ) D + D = D + D = 3 = ( ) ( ) D + D = D 3 + D + D = 7

11 Uniform hypertrees Recovering the degree sequence Not a complete invariant Below is an example of two nonisomoprhic 3-uniform hypertrees with vertices and 0 hyperedges which have the same chromatic symmetric function

12 Oriented coloring Acyclic coloring Defining generalized graphs A generalized graph on vertex set V is a collection of graphs G = {G,..., G m } where G i = (V, E i ). A map f : V P is a proper coloring if for all i there exists uv E i with f (u) f (v). Example Given any hypergraph H = (V, E) we define the generalized graph G H = {G e } e E where G e = (V, E e ) and E e = {uv : u v e} (i.e. for each hyperedge we take a clique). Proper colorings of H will correspond to proper colorings of G H and conversely.

13 Oriented coloring Acyclic coloring A hypergraph and its generalized graph 3 Figure: A hypergraph H { 3, 3} Figure: The corresponding generalized graph G H

14 Oriented coloring Acyclic coloring Oriented coloring An oriented graph G = (V, E) is an orientation on some simple graph G = (V, E). This means an oriented graph is a directed graph with no loops or opposite arcs. A tournament is an orientation of a complete graph. A proper coloring of an oriented graph G = (V, E) is a map f : V P such that: If (u, v) E, then f (u) f (v). If (u, v ), (u, v ) E, then f (u ) f (v ) or f (u ) f (v ). Notice if f : V P is a proper coloring using k colors, then f will induce a homomorphism from G to a tournament K k.

15 Oriented coloring Acyclic coloring Oriented coloring as generalized graph coloring Any instance of oriented coloring can be realized as generalized graph coloring. G = 3 G G = { 3, 3, 3} Figure: An oriented graph G and corresponding generalized graph G G

16 Oriented coloring Acyclic coloring Acyclic coloring Given a graph G = (V, E) an acyclic coloring of G is f : V P such that: The map f is a proper coloring of G. Each cycle in G uses at least three colors.

17 Oriented coloring Acyclic coloring Acyclic coloring as generalized graph coloring Any instance of acyclic coloring can be realized as generalized graph coloring. G = 3 G a,g = { 3, 3, 3, 3, 3} Figure: A graph G and corresponding generalized graph G a,g

18 Oriented coloring Acyclic coloring The end T H A N K F O R L I S T E N I N G