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Graph Saturation In Color - University Of Illinois At .

Graph Saturation In Color - University Of Illinois At .
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Graph Saturation in ColorMichael FerraraJaehoon KimElyse Yeager*yeager2@illinois.eduSan Jose State UniversityFebruary 2015Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20151 / 24

Outline1Introduction to Graphs2Graph Saturation3Ramsey Theory4Colored Graph SaturationFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20152 / 24

What Do You Know About Graphs?Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20153 / 24

Graph BasicsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph Basicsvertex G : number of vertices in GFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph Basicsvertex G : number of vertices in GFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph Basicsedgevertex G : number of vertices in GkG k : number of edges in GFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsSubgraph:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsSubgraph:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsSubgraph:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsSubgraph:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsSubgraph:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsSubgraph:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsSubgraph:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Graph BasicsSubgraph:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20154 / 24

Common GraphsPath:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20155 / 24

Common GraphsPath:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20155 / 24

Common GraphsPath:Matching:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20155 / 24

Common GraphsPath:Matching:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20155 / 24

Common GraphsPath:Matching:Clique:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20155 / 24

Common GraphsPath:Matching:Clique:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20155 / 24

Common GraphsPath:Matching:Clique:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20155 / 24

Common GraphsPath:Matching:Clique:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20155 / 24

Outline1Introduction to Graphs2Graph Saturation3Ramsey Theory4Colored Graph SaturationFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20156 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.This graph is triangle saturatedFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.This graph is triangle saturatedFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.This graph is triangle saturatedFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.DefinitionA graph G is H-saturated if:Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.DefinitionA graph G is H-saturated if:1H is not a subgraph of GFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Goal:Avoid a forbidden subgraph, but add as many edges as possible.Example:Let the triangle be forbidden.DefinitionA graph G is H-saturated if:1H is not a subgraph of G and2If we add any edge to G , H is a subgraph of the resulting graph.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20157 / 24

Saturation NumberFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20158 / 24

Saturation NumberDefinition: (Erdős-Hajnal-Moon)The saturation number of a graph H and a number n is the minimumnumber of edges in a graph on n vertices that is H saturated.sat(n; H) : min{kG k : G n and G is H saturated}Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20158 / 24

Saturation Number7 edgesDefinition: (Erdős-Hajnal-Moon)The saturation number of a graph H and a number n is the minimumnumber of edges in a graph on n vertices that is H saturated.sat(n; H) : min{kG k : G n and G is H saturated}Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20158 / 24

Saturation Number7 edges8 edgesDefinition: (Erdős-Hajnal-Moon)The saturation number of a graph H and a number n is the minimumnumber of edges in a graph on n vertices that is H saturated.sat(n; H) : min{kG k : G n and G is H saturated}Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20158 / 24

Saturation Number7 edges8 edges5 edgesDefinition: (Erdős-Hajnal-Moon)The saturation number of a graph H and a number n is the minimumnumber of edges in a graph on n vertices that is H saturated.sat(n; H) : min{kG k : G n and G is H saturated}Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20158 / 24

Saturation Number7 edges8 edges5 edgesDefinition: (Erdős-Hajnal-Moon)The saturation number of a graph H and a number n is the minimumnumber of edges in a graph on n vertices that is H saturated.sat(n; H) : min{kG k : G n and G is H saturated}Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20158 / 24

Saturation Number7 edges8 edges5 edgesDefinition: (Erdős-Hajnal-Moon)The saturation number of a graph H and a number n is the minimumnumber of edges in a graph on n vertices that is H saturated.sat(n; H) : min{kG k : G n and G is H saturated}Given the above examples:sat(6; triangle)Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20158 / 24

Saturation Number7 edges8 edges5 edgesDefinition: (Erdős-Hajnal-Moon)The saturation number of a graph H and a number n is the minimumnumber of edges in a graph on n vertices that is H saturated.sat(n; H) : min{kG k : G n and G is H saturated}Given the above examples:sat(6; triangle) 5Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20158 / 24

Outline1Introduction to Graphs2Graph Saturation3Ramsey Theory4Colored Graph SaturationFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 20159 / 24

Edge ColoringDefinition:An edge coloring of a graph is an assignment of a color to each edge.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201510 / 24

Edge ColoringDefinition:An edge coloring of a graph is an assignment of a color to each edge.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201510 / 24

Edge ColoringDefinition:An edge coloring of a graph is an assignment of a color to each edge.Colors: red, blue.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201510 / 24

Edge ColoringDefinition:An edge coloring of a graph is an assignment of a color to each edge.Colors: red, blue.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201510 / 24

Edge ColoringDefinition:An edge coloring of a graph is an assignment of a color to each edge.Colors: red, blue.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201510 / 24

Edge ColoringDefinition:An edge coloring of a graph is an assignment of a color to each edge.Definition:A graph is monochromatic if all its edges are assigned the same color.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201510 / 24

Ramsey ProblemsGoal:An edge-coloring with no forbidden monochromatic subgraph.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201511 / 24

Ramsey ProblemsGoal:An edge-coloring with no forbidden monochromatic subgraph.Example:Color red and blue; forbid red triangle and blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201511 / 24

Ramsey ProblemsGoal:An edge-coloring with no forbidden monochromatic subgraph.Example:Color red and blue; forbid red triangle and blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201511 / 24

Ramsey ProblemsGoal:An edge-coloring with no forbidden monochromatic subgraph.Example:Color red and blue; forbid red triangle and blue triangle.good coloringFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201511 / 24

Ramsey ProblemsGoal:An edge-coloring with no forbidden monochromatic subgraph.Example:Color red and blue; forbid red triangle and blue triangle.good coloringFerrara-Kim-Yeager (UCD, UIUC)bad coloringGraph Saturation in ColorFeb 201511 / 24

Ramsey ProblemsGoal:An edge-coloring with no forbidden monochromatic subgraph.Example:Color red and blue; forbid red triangle and blue triangle.good coloringFerrara-Kim-Yeager (UCD, UIUC)bad coloringGraph Saturation in Colorbad coloringFeb 201511 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redFerrara-Kim-Yeager (UCD, UIUC)or a monochromatic blueGraph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redFerrara-Kim-Yeager (UCD, UIUC)or a monochromatic blueGraph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redFerrara-Kim-Yeager (UCD, UIUC)or a monochromatic blueGraph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redFerrara-Kim-Yeager (UCD, UIUC)or a monochromatic blueGraph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redFerrara-Kim-Yeager (UCD, UIUC)or a monochromatic blueGraph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redFerrara-Kim-Yeager (UCD, UIUC)or a monochromatic blueGraph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redFerrara-Kim-Yeager (UCD, UIUC)or a monochromatic blueGraph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redFerrara-Kim-Yeager (UCD, UIUC)or a monochromatic blueGraph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K5 contains amonochromatic redor a monochromatic blueno good coloring existsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201512 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.no good coloring existsFerrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Ramsey’s TheoremGiven any collection of forbidden subgraphs, and any sufficiently largeclique, no good coloring exists.The lowest “sufficiently large” number for a given collection of subgraphsis called the Ramsey Number.Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Ramsey’s TheoremGiven any collection of forbidden subgraphs, and any sufficiently largeclique, no good coloring exists.The lowest “sufficiently large” number for a given collection of subgraphsis called the Ramsey Number.R(3, 3) 6 (two triangles)Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ramsey ProblemsNote:Sometimes a good coloring doesn’t exist!Ramsey’s TheoremGiven any collection of forbidden subgraphs, and any sufficiently largeclique, no good coloring exists.The lowest “sufficiently large” number for a given collection of subgraphsis called the Ramsey Number.R(3, 3) 6 (two triangles)R(4, 4) 18Ferrara-Kim-Yeager (UCD, UIUC)Graph Saturation in ColorFeb 201513 / 24

Ram

San Jose State University February 2015 Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 1 / 24. Outline 1 Introduction to Graphs 2 Graph Saturation 3 Ramsey Theory 4 Colored Graph Saturation Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 2 / 24. What Do You Know About Graphs?

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Recent Developments in Health and Safety Law Litigation Privilege and Internal Investigations 2 recent cases have called into question whether litigation privilege can apply to internal investigations including health and safety matters. To claim litigation privilege the following must apply: - Litigation must be in progress or in .
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