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 0/ 11
Edge coverings Goal: Relate several edge cover problems with an algebraic invariant of graphs. 1/ 11
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 . 1/ 11
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 H i with some property are needed to cover the edges of G ? 1/ 11
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 H i with some property are needed to cover the edges of G ? Such problems are fundamental in graph theory. For example: 1/ 11
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 H i 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 1/ 11
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 H i 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 . 1/ 11
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 H i 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 H i to be H − together with all incident edges, we i obtain an edge cover of G . 1/ 11
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 H i 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 H i to be H − together with all incident edges, we i obtain an edge cover of G . Ex: 1/ 11
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 H i 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 H i to be H − together with all incident edges, we i obtain an edge cover of G . Ex: Coloring 1/ 11
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 H i 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 H i to be H − together with all incident edges, we i obtain an edge cover of G . Ex: Coloring Complement graph 1/ 11
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 H i 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 H i to be H − together with all incident edges, we i obtain an edge cover of G . Ex: Coloring Complement graph H red in edge cover 1/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { col i + adj edges } 2/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { col i + adj edges } 1. (*) Split covers 2/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { 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). 2/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { 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 2/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { 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/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { 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 2/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { 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 K m , n . Tuza showed 2/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { 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 K m , n . Tuza showed biclique cover # G ≤ | V | − log 2 | V | . 2/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { 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 K m , n . Tuza showed biclique cover # G ≤ | V | − log 2 | V | . 3. Chain graph covers 2/ 11
Edge coverings: examples Problem: # graph w property required to cover edges of G . Example: Coloring G induces edge cover: H i = { 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 K m , n . Tuza showed biclique cover # G ≤ | V | − log 2 | V | . 3. Chain graph covers A chain graph is a bipartite graph w no induced 2 K 2 . 2/ 11
Recommend
More recommend
