matchings, coverings, and castelnuovo-mumford regularity - woodroofe.pdf · achain graph...

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

Matchings, coverings, andCastelnuovo-Mumford regularity

Russ WoodroofeWashington U in St [email protected]

0/ 11

Edge coverings

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

Let G be a simple graph on vertex set V and edge set E .

(Edge) covering problem: how many (not nec. induced) subgraphsHi 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 GA 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

obtain an edge cover of G .

Ex:

Coloring Complement graph Hred in edge cover

1/ 11

Edge coverings





1. Colorings of the complement graph G

A k-coloring of G divides V (G ) into cliques H−1 , . . . ,H

−k .



Ex:


1/ 11

Edge coverings






1 , . . . ,H−k .



Ex:


1/ 11

Edge coverings






1 , . . . ,H−k .



Ex:

Coloring

Complement graph Hred in edge cover

1/ 11

Edge coverings






1 , . . . ,H−k .



Ex:

Coloring Complement graph

Hred in edge cover

1/ 11

Edge coverings






1 , . . . ,H−k .



Ex:


1/ 11

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 coversA split graph is a graph whose vertices can be partitioned intoa clique and an independent set (with some edges between thetwo). We’ve seen that any coloring of G induces a covering bysplit graphs, so

split cover # G ≤ χ(G ).2. Biclique covers

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

3. Chain graph coversA 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” isNP-complete.

2/ 11



1. (*) Split covers

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




2/ 11



1. (*) Split coversA split graph is a graph whose vertices can be partitioned intoa clique and an independent set (with some edges between thetwo).

We’ve seen that any coloring of G induces a covering bysplit graphs, so




2/ 11







2/ 11




split cover # G ≤ χ(G ).

2. Biclique coversCover 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” isNP-complete.

2/ 11







2/ 11





Cover edges by bicliques Km,n. Tuza showed

biclique cover # G ≤ |V | − log2 |V |.3. Chain graph covers


2/ 11







2/ 11






3. Chain graph covers


2/ 11






3. Chain graph coversA 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” isNP-complete.

2/ 11







2/ 11


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

4. Boxicity of complement

The boxicity of G is the min # “interval graphs” that G canbe 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 intersectionproblem on G , as we’ve seen with colorings and boxicity.

3/ 11



4. Boxicity of complementThe boxicity of G is the min # “interval graphs” that G canbe written as the intersection of.

Hence, boxicity of G is the co-interval cover # of G .


3/ 11



4. Boxicity of complementThe boxicity of G is the min # “interval graphs” that G canbe written as the intersection of.Hence, boxicity of G is the co-interval cover # of G .


3/ 11

Edge coverings: co-chordal

Problem: # graph w property required to cover edges of G .Ex: Colorings -> Split graph covers, Biclique covers, Chain graphcovers, 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 everyinduced cycle has length 3.)

I split graphs are clear chordal, and the family is closed undercomplementation.

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

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

Denote as co-chordal cover # as cochordG .

4/ 11



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 everyinduced cycle has length 3.)





4/ 11





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

I Chain graphs and co-interval graphs similarly.



4/ 11








4/ 11

Table of contentsPart 1: Coverings

Part 2: Matchings

Part 3: Algebra

Matchings

cochordG = 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: cochordG ≤ Matching # G .Proof: Any maximal matching induces a split cover of G .

Proof:

(Hi is ith edge of matching + adjacent edges.)

The induced matching # of G is the largest n s.t. Mn is aninduced subgraph of G .Write as indmatchG .

Proposition: indmatchG ≤ cochordG.Proof: M2 is a 4-cycle, hence any cochordal subgraph contains atmost one edge of the induced Mn.

5/ 11

Matchings




Proposition: cochordG ≤ Matching # G .

Proof: Any maximal matching induces a split cover of G .

Proof:




5/ 11

Matchings





Proof:




5/ 11

Matchings





Proof:


The induced matching # of G is the largest n s.t. Mn is aninduced subgraph of G .

Write as indmatchG .


5/ 11

Matchings





Proof:




5/ 11

Matchings





Proof:



Proposition: indmatchG ≤ cochordG.

Proof: M2 is a 4-cycle, hence any cochordal subgraph contains atmost one edge of the induced Mn.

5/ 11

Matchings





Proof:




5/ 11

Induced matchings and cochordal covers

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

The difference cochordG − indmatchG can be arbitrarily large:cochordC5 − indmatchC5 = 1, andboth 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 indmatchG = cochordG .

(Weakly chordal ≡ every induced cycle in G and G has length ≤ 4).

6/ 11



The difference cochordG − indmatchG can be arbitrarily large:

cochordC5 − indmatchC5 = 1, andboth parameters sum over disjoint union of graphs.




6/ 11



The difference cochordG − indmatchG can be arbitrarily large:cochordC5 − indmatchC5 = 1, and

both parameters sum over disjoint union of graphs.




6/ 11







6/ 11




But for interesting classes of graphs, equality can occur.

The best result of this type that I’m aware of:



6/ 11







6/ 11

Table of contentsPart 1: CoveringsPart 2: Matchings

Part 3: Algebra

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 monomialscorresponding to independent sets of G .

The algebraic properties of the edge ring are closely connected totopological properties of the independence complex IndG .

Pseudo-definition: The Castelnuovo-Mumford regularity of theedge 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 IndG 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 independencecomplex or local cohomology of edge ring.)

7/ 11



F [G ] , F [x1, . . . , xn] / (xixj : {i , j} ∈ E ) ,






7/ 11



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 monomialscorresponding to independent sets of G .





7/ 11



F [G ] , F [x1, . . . , xn] / (xixj : {i , j} ∈ E ) ,






7/ 11



F [G ] , F [x1, . . . , xn] / (xixj : {i , j} ∈ E ) ,




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

ii. If IndG is an n-sphere, then reg F [G ] = n + 1.iii. reg(F [G1∪̇G2]) = reg F [G1] + reg F [G2].


7/ 11



F [G ] , F [x1, . . . , xn] / (xixj : {i , j} ∈ E ) ,




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

iii. reg(F [G1∪̇G2]) = reg F [G1] + reg F [G2].


7/ 11



F [G ] , F [x1, . . . , xn] / (xixj : {i , j} ∈ E ) ,






7/ 11



F [G ] , F [x1, . . . , xn] / (xixj : {i , j} ∈ E ) ,





(Real definition involves either simplicial homology of independencecomplex or local cohomology of edge ring.) 7/ 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 IndG 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 edgesSince 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 IndM2 is the square, IndM3 is the octahedron, andin general IndMn is the (n − 1)-diml “cross polytope”.

Corollary: reg F [G ] ≥ indmatchG .

8/ 11



Examples:

1. reg F [G ] = 0 ⇐⇒ G has no edges

Since Ind(edge) = S0, and so reg F [edge] = 1.



8/ 11



Examples:




8/ 11



Examples:


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

Since reg F [edge] = 1, and regularity adds over disjoint union.Or, notice IndM2 is the square, IndM3 is the octahedron, andin general IndMn is the (n − 1)-diml “cross polytope”.


8/ 11



Examples:


2. reg F [Mn] = n, where Mn is the n-matching.Since reg F [edge] = 1, and regularity adds over disjoint union.

Or, notice IndM2 is the square, IndM3 is the octahedron, andin general IndMn is the (n − 1)-diml “cross polytope”.


8/ 11



Examples:




8/ 11



3. regCn = 2, where Cn is the cyclic graph.Since IndCn 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 ] ≤ cochordG .(Follows from a deep theorem of Kalai and Meshulam.)

9/ 11



3. regCn = 2, where Cn is the cyclic graph.

Since IndCn is a circle (Cn as a 1-diml simplicial complex).




9/ 11



3. regCn = 2, where Cn is the cyclic graph.Since IndCn is a circle

(Cn as a 1-diml simplicial complex).




9/ 11







9/ 11




Corollary: reg F [G ] = 1 ⇐⇒ G is co-chordal.

Proof:( =⇒ ) is (3).(⇐=) “Tree-like” structure of chordal graphs



9/ 11




Corollary: reg F [G ] = 1 ⇐⇒ G is co-chordal.Proof:

( =⇒ ) is (3).(⇐=) “Tree-like” structure of chordal graphs



9/ 11




Corollary: reg F [G ] = 1 ⇐⇒ G is co-chordal.Proof:( =⇒ ) is (3).

(⇐=) “Tree-like” structure of chordal graphs=⇒ Ind(cochordal) is contractible.


9/ 11







9/ 11






Corollary: reg F [G ] ≤ cochordG .

(Follows from a deep theorem of Kalai and Meshulam.)

9/ 11







9/ 11

Matchings, coverings, and Castelnuovo-Mumford regularity

reg F [G ] – some algebraic invariant of G with

reg F [G ] –

indmatchG ≤ reg F [G ] ≤ cochordG .

Easy consequences:reg F [G ] ≤ matching #G

(since maximal matching induces co-chordal cover)

If G is weakly co-chordal, then reg F [G ] = indmatchG .(since Busch-Dragan-Sritharan =⇒ indmatchG = cochordG .)

Etc.

10/ 11



reg F [G ] –




If G is weakly co-chordal, then reg F [G ] = indmatchG .

(since Busch-Dragan-Sritharan =⇒ indmatchG = cochordG .)

Etc.

10/ 11



reg F [G ] –




If G is weakly co-chordal, then reg F [G ] = indmatchG .(since Busch-Dragan-Sritharan =⇒ indmatchG = cochordG .)

Etc.

10/ 11

Morals, and questions

reg F [G ] – some algebraic invariant of G withindmatchG ≤ reg F [G ] ≤ cochordG .

Moral 1: If you prove a co-chordal covering result, tell an algebraist!

Moral 2: Algebraists and algebraic combinatorialists have provedinteresting bounds on reg F [G ] with other techniques, whichsuggest graph covering results.

(Nevo) If G is claw-free with indmatchG = 1, then reg F [G ] ≤ 2.Question: If G is claw-free, is cochordG ≤ 2 · indmatchG?

(Kummini) If G is well-covered and bipartite, thenreg F [G ] = indmatchG .

Question: If G is well-covered and bipartite, iscochordG = indmatchG?

11/ 11





(Nevo) If G is claw-free with indmatchG = 1, then reg F [G ] ≤ 2.

Question: If G is claw-free, is cochordG ≤ 2 · indmatchG?



11/ 11





(Nevo) If G is claw-free with indmatchG = 1, then reg F [G ] ≤ 2.Question: If G is claw-free, is cochordG ≤ 2 · indmatchG?



11/ 11

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

Thank you!

Russ [email protected]

matchings, coverings, and castelnuovo-mumford regularity - woodroofe.pdf · achain graph...

Documents