Matchings, coverings, and Castelnuovo-Mumford regularity Russ - - PowerPoint PPT Presentation

24th Cumberland Conference on Combinatorics, Graph Theory, and Computing

Matchings, coverings, and Castelnuovo-Mumford regularity

Russ Woodroofe Washington U in St Louis russw@math.wustl.edu

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs.

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E.

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G?

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G? Such problems are fundamental in graph theory. For example:

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G? Such problems are fundamental in graph theory. For example:

• 1. Colorings of the complement graph G

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G? Such problems are fundamental in graph theory. For example:

• 1. Colorings of the complement graph G

A k-coloring of G divides V (G) into cliques H−

1 , . . . , H− k .

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G? Such problems are fundamental in graph theory. For example:

• 1. Colorings of the complement graph G

A k-coloring of G divides V (G) into cliques H−

1 , . . . , H− k .

If we take Hi to be H−

i

together with all incident edges, we

• btain an edge cover of G.

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G? Such problems are fundamental in graph theory. For example:

• 1. Colorings of the complement graph G

A k-coloring of G divides V (G) into cliques H−

1 , . . . , H− k .

If we take Hi to be H−

i

together with all incident edges, we

• btain an edge cover of G.

Ex:

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G? Such problems are fundamental in graph theory. For example:

• 1. Colorings of the complement graph G

A k-coloring of G divides V (G) into cliques H−

1 , . . . , H− k .

If we take Hi to be H−

i

together with all incident edges, we

• btain an edge cover of G.

Ex: Coloring

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G? Such problems are fundamental in graph theory. For example:

• 1. Colorings of the complement graph G

A k-coloring of G divides V (G) into cliques H−

1 , . . . , H− k .

If we take Hi to be H−

i

together with all incident edges, we

• btain an edge cover of G.

Ex: Coloring Complement graph

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Edge coverings

Goal: Relate several edge cover problems with an algebraic invariant of graphs. Let G be a simple graph on vertex set V and edge set E. (Edge) covering problem: how many (not nec. induced) subgraphs Hi with some property are needed to cover the edges of G? Such problems are fundamental in graph theory. For example:

• 1. Colorings of the complement graph G

A k-coloring of G divides V (G) into cliques H−

1 , . . . , H− k .

If we take Hi to be H−

i

together with all incident edges, we

• btain an edge cover of G.

Ex: Coloring Complement graph Hred in edge cover

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two).

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two). We’ve seen that any coloring of G induces a covering by split graphs, so

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two). We’ve seen that any coloring of G induces a covering by split graphs, so split cover # G ≤ χ(G).

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two). We’ve seen that any coloring of G induces a covering by split graphs, so split cover # G ≤ χ(G).

• 2. Biclique covers

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two). We’ve seen that any coloring of G induces a covering by split graphs, so split cover # G ≤ χ(G).

• 2. Biclique covers

Cover edges by bicliques Km,n. Tuza showed

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two). We’ve seen that any coloring of G induces a covering by split graphs, so split cover # G ≤ χ(G).

• 2. Biclique covers

Cover edges by bicliques Km,n. Tuza showed biclique cover # G ≤ |V | − log2 |V |.

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two). We’ve seen that any coloring of G induces a covering by split graphs, so split cover # G ≤ χ(G).

• 2. Biclique covers

Cover edges by bicliques Km,n. Tuza showed biclique cover # G ≤ |V | − log2 |V |.

• 3. Chain graph covers

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two). We’ve seen that any coloring of G induces a covering by split graphs, so split cover # G ≤ χ(G).

• 2. Biclique covers

Cover edges by bicliques Km,n. Tuza showed biclique cover # G ≤ |V | − log2 |V |.

• 3. Chain graph covers

A chain graph is a bipartite graph w no induced 2K2.

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Example: Coloring G induces edge cover: Hi = {col i + adj edges}

• 1. (*) Split covers

A split graph is a graph whose vertices can be partitioned into a clique and an independent set (with some edges between the two). We’ve seen that any coloring of G induces a covering by split graphs, so split cover # G ≤ χ(G).

• 2. Biclique covers

Cover edges by bicliques Km,n. Tuza showed biclique cover # G ≤ |V | − log2 |V |.

• 3. Chain graph covers

A chain graph is a bipartite graph w no induced 2K2. The chain graph cover number was studied by Yannakakis, who used it to show that checking “partial order dimension” is NP-complete.

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers

• 4. Boxicity of complement

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers

• 4. Boxicity of complement

The boxicity of G is the min # “interval graphs” that G can be written as the intersection of.

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers

• 4. Boxicity of complement

The boxicity of G is the min # “interval graphs” that G can be written as the intersection of. Hence, boxicity of G is the co-interval cover # of G.

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Edge coverings: examples

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers

• 4. Boxicity of complement

The boxicity of G is the min # “interval graphs” that G can be written as the intersection of. Hence, boxicity of G is the co-interval cover # of G. Remark: any covering problem on G has a dual intersection problem on G, as we’ve seen with colorings and boxicity.

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Edge coverings: co-chordal

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers, co Boxicity All of the preceding families of covering subgraphs share a property: their complement is chordal.

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Edge coverings: co-chordal

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers, co Boxicity All of the preceding families of covering subgraphs share a property: their complement is chordal. (A graph is chordal if every cycle has a chord, equivalently if every induced cycle has length 3.)

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Edge coverings: co-chordal

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers, co Boxicity All of the preceding families of covering subgraphs share a property: their complement is chordal. (A graph is chordal if every cycle has a chord, equivalently if every induced cycle has length 3.)

◮ split graphs are clear chordal, and the family is closed under

complementation.

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Edge coverings: co-chordal

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers, co Boxicity All of the preceding families of covering subgraphs share a property: their complement is chordal. (A graph is chordal if every cycle has a chord, equivalently if every induced cycle has length 3.)

◮ split graphs are clear chordal, and the family is closed under

complementation.

◮ complement of biclique is two cliques, which is chordal.

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Edge coverings: co-chordal

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers, co Boxicity All of the preceding families of covering subgraphs share a property: their complement is chordal. (A graph is chordal if every cycle has a chord, equivalently if every induced cycle has length 3.)

◮ split graphs are clear chordal, and the family is closed under

complementation.

◮ complement of biclique is two cliques, which is chordal. ◮ Chain graphs and co-interval graphs similarly.

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Edge coverings: co-chordal

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers, co Boxicity All of the preceding families of covering subgraphs share a property: their complement is chordal. (A graph is chordal if every cycle has a chord, equivalently if every induced cycle has length 3.)

◮ split graphs are clear chordal, and the family is closed under

complementation.

◮ complement of biclique is two cliques, which is chordal. ◮ Chain graphs and co-interval graphs similarly.

In particular, co-chordal cover # is ≤ the above cover #’s.

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Edge coverings: co-chordal

Problem: # graph w property required to cover edges of G. Ex: Colorings -> Split graph covers, Biclique covers, Chain graph covers, co Boxicity All of the preceding families of covering subgraphs share a property: their complement is chordal. (A graph is chordal if every cycle has a chord, equivalently if every induced cycle has length 3.)

◮ split graphs are clear chordal, and the family is closed under

complementation.

◮ complement of biclique is two cliques, which is chordal. ◮ Chain graphs and co-interval graphs similarly.

In particular, co-chordal cover # is ≤ the above cover #’s. Denote as co-chordal cover # as cochord G.

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Table of contents

Part 1: Coverings Part 2: Matchings Part 3: Algebra

Matchings

cochord G = min # co-chordal subgraphs to cover edges of G.

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges.

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges. The matching # of G is the largest n s.t. Mn is a subgraph of G.

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges. The matching # of G is the largest n s.t. Mn is a subgraph of G. Proposition: cochord G ≤ Matching # G.

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges. The matching # of G is the largest n s.t. Mn is a subgraph of G. Proposition: cochord G ≤ Matching # G. Proof: Any maximal matching induces a split cover of G.

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges. The matching # of G is the largest n s.t. Mn is a subgraph of G. Proposition: cochord G ≤ Matching # G. Proof: Any maximal matching induces a split cover of G. (Hi is ith edge of matching + adjacent edges.)

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges. The matching # of G is the largest n s.t. Mn is a subgraph of G. Proposition: cochord G ≤ Matching # G. Proof: Any maximal matching induces a split cover of G. (Hi is ith edge of matching + adjacent edges.) The induced matching # of G is the largest n s.t. Mn is an induced subgraph of G.

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges. The matching # of G is the largest n s.t. Mn is a subgraph of G. Proposition: cochord G ≤ Matching # G. Proof: Any maximal matching induces a split cover of G. (Hi is ith edge of matching + adjacent edges.) The induced matching # of G is the largest n s.t. Mn is an induced subgraph of G. Write as indmatch G.

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges. The matching # of G is the largest n s.t. Mn is a subgraph of G. Proposition: cochord G ≤ Matching # G. Proof: Any maximal matching induces a split cover of G. (Hi is ith edge of matching + adjacent edges.) The induced matching # of G is the largest n s.t. Mn is an induced subgraph of G. Write as indmatch G. Proposition: indmatch G ≤ cochord G.

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Matchings

cochord G = min # co-chordal subgraphs to cover edges of G. The matching graph Mn is a graph with n disjoint edges. The matching # of G is the largest n s.t. Mn is a subgraph of G. Proposition: cochord G ≤ Matching # G. Proof: Any maximal matching induces a split cover of G. (Hi is ith edge of matching + adjacent edges.) The induced matching # of G is the largest n s.t. Mn is an induced subgraph of G. Write as indmatch G. Proposition: indmatch G ≤ cochord G. Proof: M2 is a 4-cycle, hence any cochordal subgraph contains at most one edge of the induced Mn.

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Induced matchings and cochordal covers

indmatch G = max # induced matching in G. cochord G = min # co-chordal subgraphs to cover edges of G. indmatch G ≤ cochord G.

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Induced matchings and cochordal covers

indmatch G = max # induced matching in G. cochord G = min # co-chordal subgraphs to cover edges of G. indmatch G ≤ cochord G. The difference cochord G − indmatch G can be arbitrarily large:

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Induced matchings and cochordal covers

indmatch G = max # induced matching in G. cochord G = min # co-chordal subgraphs to cover edges of G. indmatch G ≤ cochord G. The difference cochord G − indmatch G can be arbitrarily large: cochord C5 − indmatch C5 = 1, and

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Induced matchings and cochordal covers

indmatch G = max # induced matching in G. cochord G = min # co-chordal subgraphs to cover edges of G. indmatch G ≤ cochord G. The difference cochord G − indmatch G can be arbitrarily large: cochord C5 − indmatch C5 = 1, and both parameters sum over disjoint union of graphs.

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Induced matchings and cochordal covers

indmatch G = max # induced matching in G. cochord G = min # co-chordal subgraphs to cover edges of G. indmatch G ≤ cochord G. The difference cochord G − indmatch G can be arbitrarily large: cochord C5 − indmatch C5 = 1, and both parameters sum over disjoint union of graphs. But for interesting classes of graphs, equality can occur.

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Induced matchings and cochordal covers

indmatch G = max # induced matching in G. cochord G = min # co-chordal subgraphs to cover edges of G. indmatch G ≤ cochord G. The difference cochord G − indmatch G can be arbitrarily large: cochord C5 − indmatch C5 = 1, and both parameters sum over disjoint union of graphs. But for interesting classes of graphs, equality can occur. The best result of this type that I’m aware of:

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Induced matchings and cochordal covers

indmatch G = max # induced matching in G. cochord G = min # co-chordal subgraphs to cover edges of G. indmatch G ≤ cochord G. The difference cochord G − indmatch G can be arbitrarily large: cochord C5 − indmatch C5 = 1, and both parameters sum over disjoint union of graphs. But for interesting classes of graphs, equality can occur. The best result of this type that I’m aware of: Theorem (Busch, Dragan, and Sritharan 2010) If G is weakly chordal, then indmatch G = cochord G.

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Induced matchings and cochordal covers

indmatch G = max # induced matching in G. cochord G = min # co-chordal subgraphs to cover edges of G. indmatch G ≤ cochord G. The difference cochord G − indmatch G can be arbitrarily large: cochord C5 − indmatch C5 = 1, and both parameters sum over disjoint union of graphs. But for interesting classes of graphs, equality can occur. The best result of this type that I’m aware of: Theorem (Busch, Dragan, and Sritharan 2010) If G is weakly chordal, then indmatch G = cochord G. (Weakly chordal ≡ every induced cycle in G and G has length ≤ 4).

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Table of contents

Part 1: Coverings Part 2: Matchings Part 3: Algebra

Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] /

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) ,

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) , where F is a field.

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) , where F is a field. I.e., we take a quotient of a polynomial ring, leaving monomials corresponding to independent sets of G.

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) , where F is a field. I.e., we take a quotient of a polynomial ring, leaving monomials corresponding to independent sets of G. The algebraic properties of the edge ring are closely connected to topological properties of the independence complex Ind G.

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) , where F is a field. I.e., we take a quotient of a polynomial ring, leaving monomials corresponding to independent sets of G. The algebraic properties of the edge ring are closely connected to topological properties of the independence complex Ind G. Pseudo-definition: The Castelnuovo-Mumford regularity of the edge ring (or independence complex) of a graph is such that:

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) , where F is a field. I.e., we take a quotient of a polynomial ring, leaving monomials corresponding to independent sets of G. The algebraic properties of the edge ring are closely connected to topological properties of the independence complex Ind G. Pseudo-definition: The Castelnuovo-Mumford regularity of the edge ring (or independence complex) of a graph is such that:

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) , where F is a field. I.e., we take a quotient of a polynomial ring, leaving monomials corresponding to independent sets of G. The algebraic properties of the edge ring are closely connected to topological properties of the independence complex Ind G. Pseudo-definition: The Castelnuovo-Mumford regularity of the edge ring (or independence complex) of a graph is such that:

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) , where F is a field. I.e., we take a quotient of a polynomial ring, leaving monomials corresponding to independent sets of G. The algebraic properties of the edge ring are closely connected to topological properties of the independence complex Ind G. Pseudo-definition: The Castelnuovo-Mumford regularity of the edge ring (or independence complex) of a graph is such that:

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

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Connection with algebra

The edge ring of a graph G = ([n], E) is F[G] F [x1, . . . , xn] / (xixj : {i, j} ∈ E) , where F is a field. I.e., we take a quotient of a polynomial ring, leaving monomials corresponding to independent sets of G. The algebraic properties of the edge ring are closely connected to topological properties of the independence complex Ind G. Pseudo-definition: The Castelnuovo-Mumford regularity of the edge ring (or independence complex) of a graph is such that:

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2]. (Real definition involves either simplicial homology of independence complex or local cohomology of edge ring.)

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Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2]. Examples:

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Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2]. Examples:

• 1. reg F[G] = 0 ⇐

⇒ G has no edges

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Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2]. Examples:

• 1. reg F[G] = 0 ⇐

⇒ G has no edges Since Ind(edge) = S0, and so reg F[edge] = 1.

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Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2]. Examples:

• 1. reg F[G] = 0 ⇐

⇒ G has no edges Since Ind(edge) = S0, and so reg F[edge] = 1.

• 2. reg F[Mn] = n, where Mn is the n-matching.

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Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2]. Examples:

• 1. reg F[G] = 0 ⇐

⇒ G has no edges Since Ind(edge) = S0, and so reg F[edge] = 1.

• 2. reg F[Mn] = n, where Mn is the n-matching.

Since reg F[edge] = 1, and regularity adds over disjoint union.

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Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2]. Examples:

• 1. reg F[G] = 0 ⇐

⇒ G has no edges Since Ind(edge) = S0, and so reg F[edge] = 1.

• 2. reg F[Mn] = n, where Mn is the n-matching.

Since reg F[edge] = 1, and regularity adds over disjoint union. Or, notice Ind M2 is the square, Ind M3 is the octahedron, and in general Ind Mn is the (n − 1)-diml “cross polytope”.

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Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2]. Examples:

• 1. reg F[G] = 0 ⇐

⇒ G has no edges Since Ind(edge) = S0, and so reg F[edge] = 1.

• 2. reg F[Mn] = n, where Mn is the n-matching.

Since reg F[edge] = 1, and regularity adds over disjoint union. Or, notice Ind M2 is the square, Ind M3 is the octahedron, and in general Ind Mn is the (n − 1)-diml “cross polytope”. Corollary: reg F[G] ≥ indmatch G.

8/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

Since Ind C n is a circle

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

Since Ind C n is a circle (Cn as a 1-diml simplicial complex).

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

Since Ind C n is a circle (Cn as a 1-diml simplicial complex). Corollary: reg F[G] = 1 ⇐ ⇒ G is co-chordal.

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

Since Ind C n is a circle (Cn as a 1-diml simplicial complex). Corollary: reg F[G] = 1 ⇐ ⇒ G is co-chordal. Proof:

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

Since Ind C n is a circle (Cn as a 1-diml simplicial complex). Corollary: reg F[G] = 1 ⇐ ⇒ G is co-chordal. Proof: ( = ⇒ ) is (3).

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

Since Ind C n is a circle (Cn as a 1-diml simplicial complex). Corollary: reg F[G] = 1 ⇐ ⇒ G is co-chordal. Proof: ( = ⇒ ) is (3). (⇐ =) “Tree-like” structure of chordal graphs = ⇒ Ind(cochordal) is contractible.

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

Since Ind C n is a circle (Cn as a 1-diml simplicial complex). Corollary: reg F[G] = 1 ⇐ ⇒ G is co-chordal. Proof: ( = ⇒ ) is (3). (⇐ =) “Tree-like” structure of chordal graphs = ⇒ Ind(cochordal) is contractible. Corollary: reg F[G] ≤ cochord G.

9/ 11

Castelnuovo-Mumford regularity examples

Pseudo-defn: Castelnuovo-Mumford regularity reg F[G] is s.t.

• i. If H is an induced subgraph of G, then reg F[H] ≤ reg F[G].
• ii. If Ind G is an n-sphere, then reg F[G] = n + 1.
• iii. reg(F[G1 ˙

∪G2]) = reg F[G1] + reg F[G2].

• 3. reg C n = 2, where Cn is the cyclic graph.

Since Ind C n is a circle (Cn as a 1-diml simplicial complex). Corollary: reg F[G] = 1 ⇐ ⇒ G is co-chordal. Proof: ( = ⇒ ) is (3). (⇐ =) “Tree-like” structure of chordal graphs = ⇒ Ind(cochordal) is contractible. Corollary: reg F[G] ≤ cochord G. (Follows from a deep theorem of Kalai and Meshulam.)

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Matchings, coverings, and Castelnuovo-Mumford regularity

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Easy consequences: reg F[G] ≤ matching #G

10/ 11

Matchings, coverings, and Castelnuovo-Mumford regularity

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Easy consequences: reg F[G] ≤ matching #G (since maximal matching induces co-chordal cover)

10/ 11

Matchings, coverings, and Castelnuovo-Mumford regularity

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Easy consequences: reg F[G] ≤ matching #G (since maximal matching induces co-chordal cover) If G is weakly co-chordal, then reg F[G] = indmatch G.

10/ 11

Matchings, coverings, and Castelnuovo-Mumford regularity

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Easy consequences: reg F[G] ≤ matching #G (since maximal matching induces co-chordal cover) If G is weakly co-chordal, then reg F[G] = indmatch G. (since Busch-Dragan-Sritharan = ⇒ indmatch G = cochord G.)

10/ 11

Matchings, coverings, and Castelnuovo-Mumford regularity

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Easy consequences: reg F[G] ≤ matching #G (since maximal matching induces co-chordal cover) If G is weakly co-chordal, then reg F[G] = indmatch G. (since Busch-Dragan-Sritharan = ⇒ indmatch G = cochord G.) Etc.

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Morals, and questions

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G.

11/ 11

Morals, and questions

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Moral 1: If you prove a co-chordal covering result, tell an algebraist!

11/ 11

Morals, and questions

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Moral 1: If you prove a co-chordal covering result, tell an algebraist! Moral 2: Algebraists and algebraic combinatorialists have proved interesting bounds on reg F[G] with other techniques, which suggest graph covering results.

11/ 11

Morals, and questions

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Moral 1: If you prove a co-chordal covering result, tell an algebraist! Moral 2: Algebraists and algebraic combinatorialists have proved interesting bounds on reg F[G] with other techniques, which suggest graph covering results. (Nevo) If G is claw-free with indmatch G = 1, then reg F[G] ≤ 2.

11/ 11

Morals, and questions

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Moral 1: If you prove a co-chordal covering result, tell an algebraist! Moral 2: Algebraists and algebraic combinatorialists have proved interesting bounds on reg F[G] with other techniques, which suggest graph covering results. (Nevo) If G is claw-free with indmatch G = 1, then reg F[G] ≤ 2. Question: If G is claw-free, is cochord G ≤ 2 · indmatch G?

11/ 11

Morals, and questions

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Moral 1: If you prove a co-chordal covering result, tell an algebraist! Moral 2: Algebraists and algebraic combinatorialists have proved interesting bounds on reg F[G] with other techniques, which suggest graph covering results. (Nevo) If G is claw-free with indmatch G = 1, then reg F[G] ≤ 2. Question: If G is claw-free, is cochord G ≤ 2 · indmatch G? (Kummini) If G is well-covered and bipartite, then reg F[G] = indmatch G.

11/ 11

Morals, and questions

reg F[G] – some algebraic invariant of G with indmatch G ≤ reg F[G] ≤ cochord G. Moral 1: If you prove a co-chordal covering result, tell an algebraist! Moral 2: Algebraists and algebraic combinatorialists have proved interesting bounds on reg F[G] with other techniques, which suggest graph covering results. (Nevo) If G is claw-free with indmatch G = 1, then reg F[G] ≤ 2. Question: If G is claw-free, is cochord G ≤ 2 · indmatch G? (Kummini) If G is well-covered and bipartite, then reg F[G] = indmatch G. Question: If G is well-covered and bipartite, is cochord G = indmatch G?

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Reference: Russ Woodroofe, Matchings, coverings, and Castelnuovo-Mumford regularity, arXiv:1009.2756.

Thank you!

Russ Woodroofe russw@math.wustl.edu