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24th Cumberland Conference on Combinatorics, Graph Theory, and Computing
Matchings, coverings, andCastelnuovo-Mumford regularity
Russ WoodroofeWashington U in St Louisrussw@math.wustl.edu
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
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
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
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
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 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
obtain an edge cover of G .
Ex:
Coloring Complement graph Hred in edge cover
1/ 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
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
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
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
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
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: 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
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 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
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
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
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 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
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
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 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
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
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 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
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
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
Edge coverings: examples
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
Edge coverings: examples
Problem: # graph w property required to cover edges of G .Ex: Colorings -> Split graph covers, Biclique covers, Chain graphcovers
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 .
Remark: any covering problem on G has a dual intersectionproblem on G , as we’ve seen with colorings and boxicity.
3/ 11
Edge coverings: examples
Problem: # graph w property required to cover edges of G .Ex: Colorings -> Split graph covers, Biclique covers, Chain graphcovers
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 .
Remark: any covering problem on G has a dual intersectionproblem on G , as we’ve seen with colorings and boxicity.
3/ 11
Edge coverings: examples
Problem: # graph w property required to cover edges of G .Ex: Colorings -> Split graph covers, Biclique covers, Chain graphcovers
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 .
Remark: any covering problem on G has a dual intersectionproblem on G , as we’ve seen with colorings and boxicity.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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, 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 indmatchG = cochordG .
(Weakly chordal ≡ every induced cycle in G and G has length ≤ 4).
6/ 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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 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
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
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
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
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
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
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].
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
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].
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
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].
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
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].
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
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].
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
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].
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
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].
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
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].
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
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].
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
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].
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
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
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
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
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
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
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
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
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
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
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
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
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
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