A Deletion-Contraction Relation for theChromatic Symmetric Function
Logan Crew (Penn), Sophie Spirkl (Princeton)
University of Albany Discrete Math 2-Day
April 25-26, 2020
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 1 / 16
A Graph
Atlanta
MontgomerySan Antonio
Minneapolis St. Louis
The Graph ∆ = (V,E): Airports and Flights
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 2 / 16
Edge Deletion and Contraction in Graphs
Atlanta
Montgomery
San Antonio
Minneapolis St. Louis
Atlanta
MontgomerySan Antonio
Minneapolis St. Louis
Deletion: ∆\(Min-San)
Atlanta
Montgomery
Min Antonio
St. Louis
Contraction: ∆/(Min-San)
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 3 / 16
A Deletion-Contraction Relation for χG
Definition (Birkhoff)
The chromatic polynomial χG(x) is defined by letting χG(n) be thenumber of n-colorings of G for all n ∈ N.
Theorem (Folklore)
For every graph G = (V,E) and any edge e ∈ E,
χG(x) = χG\e(x)− χG/e(x).
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 4 / 16
The Chromatic Symmetric Function
Let G = (V,E) be a graph.
Definition (Stanley (1995))
XG(x1, x2, . . . ) =∑col. κ
∏v∈V
xκ(v)
This function is a power series in R[[x1, x2, . . . ]]. It is called a symmetricfunction because for every permutation π of N,
f(x1, x2, . . . ) = f(xπ(1), xπ(2), . . . ).
This function is a generalization of the chromatic polynomial since
XG(1, 1, . . . , 1︸ ︷︷ ︸n 1s
, 0, 0, . . . ) = χG(n)
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 5 / 16
The Chromatic Symmetric Function
Let G = (V,E) be a graph.
Definition (Stanley (1995))
XG(x1, x2, . . . ) =∑col. κ
∏v∈V
xκ(v)
This function is a power series in R[[x1, x2, . . . ]]. It is called a symmetricfunction because for every permutation π of N,
f(x1, x2, . . . ) = f(xπ(1), xπ(2), . . . ).
This function is a generalization of the chromatic polynomial since
XG(1, 1, . . . , 1︸ ︷︷ ︸n 1s
, 0, 0, . . . ) = χG(n)
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 5 / 16
Computing X∆
Let blue = 1, green = 2, red = 3.
X∆ = x31x2x3 + · · ·+ x2
1x22x3 + . . .
Atlanta
Montgomery
San Antonio
Minneapolis St. Louis
x31x2x3
Atlanta
Montgomery
San Antonio
Minneapolis St. Louis
x21x22x3
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 6 / 16
Vertex-Weighted XG
Let G = (V,E) be a graph.
Definition (Stanley (1995))
XG(x1, x2, . . . ) =∑col. κ
∏v∈V
xκ(v)
Let w : V → N.
Definition (C.-Spirkl (2019))
X(G,w)(x1, x2, . . . ) =∑col. κ
∏v∈V
xw(v)κ(v)
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 7 / 16
Vertex-Weighted XG
Let G = (V,E) be a graph.
Definition (Stanley (1995))
XG(x1, x2, . . . ) =∑col. κ
∏v∈V
xκ(v)
Let w : V → N.
Definition (C.-Spirkl (2019))
X(G,w)(x1, x2, . . . ) =∑col. κ
∏v∈V
xw(v)κ(v)
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 7 / 16
A Deletion-Contraction Relation
I X(G,w)(1, 1, . . . , 1︸ ︷︷ ︸n 1s
, 0, 0, . . . ) = χG(n)
I X(G,w) is homogeneous of degree∑
v∈V w(v)
Theorem (C.-Spirkl (2019))
Let (G,w) be a vertex-weighted graph, and let e be any edge of G. Then
X(G,w) = X(G\e,w) −X(G/e,w/e).
Here w/e means that when the edge e is contracted, the weights of thecontracted vertices are added.
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 8 / 16
A Deletion-Contraction Relation
1
12
3 2
1
12
3 2
1
1
5
2
X(G\e,w) = X(G,w) +X(G/e,w/e)
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 9 / 16
A Deletion-Contraction Relation
1
12
3 2
1
12
3 2
1
1
5
2
X(G\e,w) = X(G,w) +X(G/e,w/e)
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 10 / 16
Acyclic Orientations
In a directed graph G, a sink is a vertex with no out-edges.
Theorem (Stanley (1995))
Let G = (V,E), and let XG =∑cλeλ, where {eλ} is the basis of
elementary symmetric functions. Then the number of acyclic orientationsof G is ∑
cλ.
The number of acyclic orientations of G with exactly k sinks is∑λ hask parts
cλ.
I Analogue of the formula (−1)mχG(−1) for acyclic orientations
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 11 / 16
Acyclic Orientations
For an acyclic orientation γ of (G,w), let Sink(γ) be the set of sinkvertices, and sink(γ) = |Sink(γ)|.
Define a sink map of γ to be a map S : V → 2N such that S(v) ⊆ [w(v)]and S(v) 6= ∅ iff v ∈ Sink(γ).
Theorem (C.-Spirkl (2019))
Let n = |V |, d =∑
v∈V w(v), and X(G,w) =∑
λ`d cλeλ. Then∑λ hask parts
cλ = (−1)d−n∑(γ,S)
(−1)k−sink(γ)
where the sum is over (γ, S) such that γ is an acyclic orientation of G, Sis a sink map of γ, and sw(G, γ, S) =
∑v∈Sink(γ) |S(v)| = k.
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 12 / 16
Acyclic Orientations: Proof Sketch
I Induction on |E|; want to show∑sw(G\e,γ,S)=k
(−1)sink(γ) =∑
sw(G,γ,S)=k
(−1)sink(γ) −∑
sw(G/e,γ,S)=k
(−1)sink(γ).
I Fix γ0, an acyclic orientation of G\e.
I Fix S0 : V → 2N with S0(v) ⊆ [w(v)] for all v.
I Get two (γ, S) for G (both orientations of e), and one (γ, S) in G/e(with S(v∗) = S(v1) ∪ {w(v1) + i : i ∈ S(v2)}).
I Only count (γ, S) if γ is acyclic, and S is a sink map for γ.
I Want to show: LHS = RHS for terms arising from γ0 and S0.
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 13 / 16
Acyclic Orientations: Proof SketchI Here: k = 3.
Case 1: There is a directed path between the endpoints of e. Thenregardless of the map S0, contraction fails, and one orientation of adding efails. The other is valid with S0 if and only if the original on G\e is.
2, ∅
1, ∅2, ∅
1, ∅ 3, {1, 2, 3}
Valid term for ∆\e with 1 sink
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof SketchI Here: k = 3.
Case 1: There is a directed path between the endpoints of e. Thenregardless of the map S0, contraction fails, and one orientation of adding efails. The other is valid with S0 if and only if the original on G\e is.
2, ∅
1, ∅2, ∅
1, ∅ 3, {1, 2, 3}
Invalid first term for ∆
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof SketchI Here: k = 3.
Case 1: There is a directed path between the endpoints of e. Thenregardless of the map S0, contraction fails, and one orientation of adding efails. The other is valid with S0 if and only if the original on G\e is.
2, ∅
1, ∅2, ∅
1, ∅ 3, {1, 2, 3}
Valid second term for ∆ with 1 sink
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof SketchI Here: k = 3.
Case 1: There is a directed path between the endpoints of e. Thenregardless of the map S0, contraction fails, and one orientation of adding efails. The other is valid with S0 if and only if the original on G\e is.
2, ∅
1, ∅
3, {1, 2, 3}
3, ∅
Invalid term for ∆/e
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
We now divide into cases based on whether one, both, or neither of theendpoints of e is a sink with respect to γ. All of these cases have fairlysimilar approaches, so we will go through just one of them, the case inwhich exactly one endpoint is a sink.
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
Subcase: S0(San Antonio) is empty (must have S0(Minneapolis) empty).
2, ∅
1, {1}2, ∅
1, ∅ 3, {1, 3}
Invalid term for ∆\e
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
Subcase: S0(San Antonio) is empty (must have S0(Minneapolis) empty).
2, ∅
1, {1}2, ∅
1, ∅ 3, {1, 3}
Invalid first term for ∆
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
Subcase: S0(San Antonio) is empty (must have S0(Minneapolis) empty).
2, ∅
1, {1}2, ∅
1, ∅ 3, {1, 3}
Valid second term for ∆ with 2 sinks
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
Subcase: S0(San Antonio) is empty (must have S0(Minneapolis) empty).
2, ∅
1, {1}
3, ∅
3, {1, 3}
Valid term for ∆/e with 2 sinks
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
Subcase: S0(San Antonio) is nonempty (must have S0(Minn.) empty).
2, ∅
1, {1}2, {2}
1, ∅ 3, {1}
Valid term for ∆\e with 3 sinks
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
Subcase: S0(San Antonio) is nonempty (must have S0(Minn.) empty).
2, ∅
1, {1}2, {2}
1, ∅ 3, {1}
Valid first term for ∆ with 3 sinks
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
Subcase: S0(San Antonio) is nonempty (must have S0(Minn.) empty).
2, ∅
1, {1}2, {2}
1, ∅ 3, {1}
Invalid second term for ∆
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Acyclic Orientations: Proof Sketch
I Here: k = 3.
Subcase: S0(San Antonio) is nonempty (must have S0(Minn.) empty).
2, ∅
1, {1}
3, {3}
3, {1}
Invalid term for ∆/e
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 14 / 16
Other Results
Let (G,w) be a vertex-weighted graph with n vertices and total weight d.
Theorem (Stanley (1995), C.-Spirkl(2019))
X(G,w) =∑
S⊆E(G)
(−1)|S|pλ(G,w,S)
where λ(G,w, S) is the partition of the total weights of the connectedcomponents of (V, S).
Theorem (Stanley (1995), C.-Spirkl(2019))∑(γ,κ)
u→γv =⇒ κ(u)≤κ(v)
∏v∈V (G)
xw(v)κ(v) = (−1)d−nω(X(G,w))
where the sum ranges over all acyclic orientations γ of G and κ is a (notnecessarily proper) coloring of G.
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 15 / 16
The End
This talk is based on the paper “A Deletion-Contraction Relation for theChromatic Symmetric Function” joint with Sophie Spirkl,https://arxiv.org/abs/1910.11859.
Thank you!
Logan Crew XG on Vertex-Weighted Graphs April 25-26, 2020 16 / 16
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