the coloring problem in graphs with structural
play

The coloring problem in graphs with structural restrictions Lucas - PowerPoint PPT Presentation

Aug 28, 2022 •254 likes •1.52k views

The coloring problem in graphs with structural restrictions Lucas Pastor Joint-work with Frdric Maffray G-SCOP March 4, 2016 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 1 / 36 The k -coloring problem k -coloring For

  1. Tools Lemma It is sufficient to produce a 4-coloring for any ( P 6 , bull)-free graph G that satisfies the following properties: 1 G and G are connected. 2 G is quasi-prime. 3 G is K 5 -free and double-wheel-free. Warning We do not claim that K 5 and the double-wheel are the only 5-vertex-critical graphs! Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 12 / 36

  2. Tools Proof of 1 If G is not connected we can examine each component of G separately. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 13 / 36

  3. Tools Proof of 1 If G is not connected we can examine each component of G separately. If G is not connected, V ( G ) can be partitioned into two non-empty sets V 1 and V 2 complete to each other. We have χ ( G ) = χ ( G [ V 1 ]) + χ ( G [ V 2 ]). Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 13 / 36

  4. Tools Proof of 1 If G is not connected we can examine each component of G separately. If G is not connected, V ( G ) can be partitioned into two non-empty sets V 1 and V 2 complete to each other. We have χ ( G ) = χ ( G [ V 1 ]) + χ ( G [ V 2 ]). ◮ We need G [ V 1 ] and G [ V 2 ] to be 3-colorable. We use the known algorithms for that. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 13 / 36

  5. Tools Proof of 1 If G is not connected we can examine each component of G separately. If G is not connected, V ( G ) can be partitioned into two non-empty sets V 1 and V 2 complete to each other. We have χ ( G ) = χ ( G [ V 1 ]) + χ ( G [ V 2 ]). ◮ We need G [ V 1 ] and G [ V 2 ] to be 3-colorable. We use the known algorithms for that. ◮ If they are 3-colorable, we can precisely determine their chromatic number by testing if they are bipartite or edgeless. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 13 / 36

  6. Tools Proof of 2 Assume that G is not quasi-prime. So it has a homogeneous set X that is not a clique. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 14 / 36

  7. Tools Proof of 2 Assume that G is not quasi-prime. So it has a homogeneous set X that is not a clique. A necessary condition for G to be 4-colorable is that χ ( G [ X ]) ≤ 3. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 14 / 36

  8. Tools Proof of 2 Assume that G is not quasi-prime. So it has a homogeneous set X that is not a clique. A necessary condition for G to be 4-colorable is that χ ( G [ X ]) ≤ 3. If it is 3-colorable, we determine χ ( G [ X ]) by testing if it is bipartite or edgeless. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 14 / 36

  9. Tools Proof of 2 Assume that G is not quasi-prime. So it has a homogeneous set X that is not a clique. A necessary condition for G to be 4-colorable is that χ ( G [ X ]) ≤ 3. If it is 3-colorable, we determine χ ( G [ X ]) by testing if it is bipartite or edgeless. Contract X into a clique of size χ ( G [ X ]). Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 14 / 36

  10. Tools Proof of 2 Assume that G is not quasi-prime. So it has a homogeneous set X that is not a clique. A necessary condition for G to be 4-colorable is that χ ( G [ X ]) ≤ 3. If it is 3-colorable, we determine χ ( G [ X ]) by testing if it is bipartite or edgeless. Contract X into a clique of size χ ( G [ X ]). Repeat this until we obtain a quasi-prime graph G ′ . Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 14 / 36

  11. Tools Proof of 2 Assume that G is not quasi-prime. So it has a homogeneous set X that is not a clique. A necessary condition for G to be 4-colorable is that χ ( G [ X ]) ≤ 3. If it is 3-colorable, we determine χ ( G [ X ]) by testing if it is bipartite or edgeless. Contract X into a clique of size χ ( G [ X ]). Repeat this until we obtain a quasi-prime graph G ′ . We can show that G is 4-colorable if and only if G ′ is 4-colorable. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 14 / 36

  12. Tools Quasi-prime Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 15 / 36

  13. Tools Quasi-prime Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 15 / 36

  14. Tools Quasi-prime Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 15 / 36

  15. Tools Proof of 3 It is easy to check that K 5 and the double-wheel are not 4-colorable. Hence, if G contains a K 5 or the double-wheel it is not 4-colorable. K 5 double-wheel Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 16 / 36

  16. Tools Structural Lemma Let G be a quasi-prime bull-free graph that contains no K 5 and no double-wheel. Then at least one of the following holds: Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 17 / 36

  17. Tools Structural Lemma Let G be a quasi-prime bull-free graph that contains no K 5 and no double-wheel. Then at least one of the following holds: G contains a magnet. special graph Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 17 / 36

  18. Tools Structural Lemma Let G be a quasi-prime bull-free graph that contains no K 5 and no double-wheel. Then at least one of the following holds: G contains a magnet. G contains a gem and a special graph. special graph Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 17 / 36

  19. Tools Structural Lemma Let G be a quasi-prime bull-free graph that contains no K 5 and no double-wheel. Then at least one of the following holds: G contains a magnet. G contains a gem and a special graph. G is gem-free. special graph Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 17 / 36

  20. Tools Magnet A subgraph F of G is a magnet if every vertex of G \ F has two neighbors u , v ∈ V ( F ) such that uv ∈ E ( F ). Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 18 / 36

  21. Tools Magnet A subgraph F of G is a magnet if every vertex of G \ F has two neighbors u , v ∈ V ( F ) such that uv ∈ E ( F ). How to use it? We can fix a coloring on F and use the 2-list-coloring algorithms to try extend it to the rest of the graph in polynomial time. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 18 / 36

  22. Tools List-coloring problem Every vertex v of the graph G has a list L ( v ) of admissible colors. We want to find a proper coloring such that c ( v ) ∈ L ( v ), for every vertex v ∈ V ( G ). Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 19 / 36

  23. Tools List-coloring problem Every vertex v of the graph G has a list L ( v ) of admissible colors. We want to find a proper coloring such that c ( v ) ∈ L ( v ), for every vertex v ∈ V ( G ). 2-list coloring List-coloring problem where all lists are of size 2. The 2-list coloring problem can be solved in polynomial time. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 19 / 36

  24. Tools Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  25. Tools v 1 {• , •} Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  26. Tools v 1 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  27. Tools v 2 {• , •} v 1 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  28. Tools v 2 v 1 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  29. Tools v 2 v 1 v 3 {• , •} Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  30. Tools v 2 v 1 v 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  31. Tools v 2 v 1 v 3 v 4 {• , •} Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  32. Tools {• , •} v 1 {• , •} {• , •} {• , •} Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  33. Tools {• , •} v 1 {• , •} {• , •} Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  34. Tools {• , •} {• , •} {• , •} Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 20 / 36

  35. Tools Magnet coloring Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 21 / 36

  36. Tools Magnet coloring Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 21 / 36

  37. Tools Magnet coloring Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 21 / 36

  38. Tools Magnet coloring Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 21 / 36

  39. Tools Magnet coloring 2-list-coloring problem Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 21 / 36

  40. Tools F 0 F 1 F 2 F 3 F 4 F 5 F 6 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 22 / 36

  41. Gem-free case Theorem [ Maffray, P. ] For any fixed k , there is a polynomial algorithm that determines if a ( P 6 , bull, gem)-free graph is k -colorable and if it is, produces a k -coloring. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 23 / 36

  42. Gem-free case Theorem [ Maffray, P. ] For any fixed k , there is a polynomial algorithm that determines if a ( P 6 , bull, gem)-free graph is k -colorable and if it is, produces a k -coloring. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 23 / 36

  43. Gem-free case Perfect graph A graph G is perfect if every induced subgraph H of G satisfies χ ( H ) = ω ( H ). Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 24 / 36

  44. Gem-free case Perfect graph A graph G is perfect if every induced subgraph H of G satisfies χ ( H ) = ω ( H ). Strong Perfect Graph Theorem [ Chudnovsky, Robertson, Seymour and Thomas 2006 ] A graph is perfect if and only if it contains no C ℓ and no C ℓ for any odd ℓ ≥ 5 . Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 24 / 36

  45. Gem-free case Coloring ( P 6 , bull, gem)-free graphs Since G is P 6 -free, it contains no C ℓ with ℓ ≥ 7, and since it is gem-free, it contains no C ℓ with ℓ ≥ 7. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 25 / 36

  46. Gem-free case Coloring ( P 6 , bull, gem)-free graphs Since G is P 6 -free, it contains no C ℓ with ℓ ≥ 7, and since it is gem-free, it contains no C ℓ with ℓ ≥ 7. If G contains no C 5 , then it is bull-free perfect. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 25 / 36

  47. Gem-free case Coloring ( P 6 , bull, gem)-free graphs Since G is P 6 -free, it contains no C ℓ with ℓ ≥ 7, and since it is gem-free, it contains no C ℓ with ℓ ≥ 7. If G contains no C 5 , then it is bull-free perfect. If G contains a C 5 , we prove that it is triangle-free. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 25 / 36

  48. Gem-free case U 1 U 5 U 2 U 4 U 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 26 / 36

  49. Gem-free case U 1 U i is anticomplete to U i − 2 ∪ U i +2 U 5 U 2 U 4 U 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 26 / 36

  50. Gem-free case U 1 U i is anticomplete to U i − 2 ∪ U i +2 U i contains a vertex that is complete to U i − 1 ∪ U i +1 U 5 U 2 U 4 U 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 26 / 36

  51. Gem-free case U 1 U i is anticomplete to U i − 2 ∪ U i +2 U i contains a vertex that is complete to U i − 1 ∪ U i +1 Each of U 1 , . . . , U 5 is a stable set U 5 U 2 U 4 U 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 26 / 36

  52. Gem-free case U 1 U i is anticomplete to U i − 2 ∪ U i +2 U i contains a vertex that is complete to U i − 1 ∪ U i +1 Each of U 1 , . . . , U 5 is a stable set There is no blue triangle U 5 U 2 U 4 U 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 26 / 36

  53. Gem-free case U 1 U i is anticomplete to U i − 2 ∪ U i +2 U i contains a vertex that is complete to U i − 1 ∪ U i +1 Each of U 1 , . . . , U 5 is a stable set There is no blue triangle U 5 U 2 There is no red triangle U 4 U 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 26 / 36

  54. Gem-free case U 1 U i is anticomplete to U i − 2 ∪ U i +2 U i contains a vertex that is complete to U i − 1 ∪ U i +1 Each of U 1 , . . . , U 5 is a stable set There is no blue triangle U 5 U 2 There is no red triangle There is no green triangle U 4 U 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 26 / 36

  55. Gem-free case U 1 U i is anticomplete to U i − 2 ∪ U i +2 U i contains a vertex that is complete to U i − 1 ∪ U i +1 Each of U 1 , . . . , U 5 is a stable set There is no blue triangle U 5 U 2 There is no red triangle There is no green triangle Hence, there is no triangle U 4 U 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 26 / 36

  56. Gem-free case Theorem [ Brandstädt et al. 2006 ] For any fixed k , k -coloring a ( P 6 , K 3 )-free graph is polynomial solvable. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 27 / 36

  57. Gem-free case Theorem [ Brandstädt et al. 2006 ] For any fixed k , k -coloring a ( P 6 , K 3 )-free graph is polynomial solvable. Lemma Let G be a prime ( P 6 , bull, gem)-free graph that contains a 5-hole. Then G is triangle-free. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 27 / 36

  58. Gem case [ Maffray, P. ] There is a polynomial time algorithm that determines whether a ( P 6 , bull)-free graphs is 4-colorable, and if it is, produces a 4-coloring. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 28 / 36

  59. Gem case [ Maffray, P. ] There is a polynomial time algorithm that determines whether a ( P 6 , bull)-free graphs is 4-colorable, and if it is, produces a 4-coloring. Structural Lemma Let G be a quasi-prime bull-free graph that contains no K 5 and no double-wheel. Then at least one of the following holds: G contains a magnet. G contains a gem and a special graph. G is gem-free. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 28 / 36

  60. Gem case [ Maffray, P. ] There is a polynomial time algorithm that determines whether a ( P 6 , bull)-free graphs is 4-colorable, and if it is, produces a 4-coloring. Structural Lemma Let G be a quasi-prime bull-free graph that contains no K 5 and no double-wheel. Then at least one of the following holds: G contains a magnet. G contains a gem and a special graph. G is gem-free. Structure By the structural Lemma and further examining the structure we can partition the graph into sets as in the next drawing. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 28 / 36

  61. Gem case How to use this structure? We will see that we can use this structure to color all the graph. In fact we color a fixed number of vertices and try to extend it. Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 29 / 36

  62. Gem case Z 1 Z 0 X W V 1 V 5 V 4 V 2 V 3 Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 30 / 36

  63. Gem case Z 1 Z 0 X W V 1 V 5 V 4 V 2 V 3 induced gem Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 30 / 36

  64. Gem case Z 1 Z 0 X W V 1 V 5 V 4 V 2 V 3 induced special graph Lucas Pastor (G-SCOP) Coloring with restrictions March 4, 2016 30 / 36

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Vertex-coloring problem The Vertex coloring problem and bipartite graphs Tyler Moore CSE 3353,

Vertex-coloring problem The Vertex coloring problem and bipartite graphs Tyler Moore CSE 3353,

Vertex-coloring problem The Vertex coloring problem and bipartite graphs Tyler Moore CSE 3353, SMU, Dallas, TX Lecture 8 Some slides created by or adapted from Dr. Kevin Wayne. For more information see

117 views • 8 slides
Vertex-coloring problem The Vertex coloring problem and bipartite graphs Tyler Moore CSE 3353,

Vertex-coloring problem The Vertex coloring problem and bipartite graphs Tyler Moore CSE 3353,

Vertex-coloring problem The Vertex coloring problem and bipartite graphs Tyler Moore CSE 3353, SMU, Dallas, TX Lecture 8 Some slides created by or adapted from Dr. Kevin Wayne. For more information see

53 views • 4 slides
Graph coloring Kevin Ly September 20, 2019 Itinerary Introduction to simple graphs A

Graph coloring Kevin Ly September 20, 2019 Itinerary Introduction to simple graphs A

Graph coloring Kevin Ly September 20, 2019 Itinerary Introduction to simple graphs A brief game of matching The coloring problem and DSATUR Networkx demo Graphs primer A simple graph G consists of a set V and a set of pairs E .

524 views • 23 slides
Obstructions against 3-coloring graphs without long induced paths Oliver Schaudt Universit at

Obstructions against 3-coloring graphs without long induced paths Oliver Schaudt Universit at

Obstructions against 3-coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH Aachen with Maria Chudnovsky, Jan Goedgebeur, and Mingxian Zhong Graph coloring Graph coloring a k-coloring is an

588 views • 45 slides
Today. Graph Coloring. Planar graphs and maps. Given G = ( V , E ) , a coloring of a G assigns

Today. Graph Coloring. Planar graphs and maps. Given G = ( V , E ) , a coloring of a G assigns

Today. Graph Coloring. Planar graphs and maps. Given G = ( V , E ) , a coloring of a G assigns colors to vertices V Planar graph coloring map coloring. where for each edge the endpoints have different colors. Planar Five Color theorem.

540 views • 4 slides
3-coloring graphs without induced paths on seven vertices Oliver Schaudt Universit at zu K

3-coloring graphs without induced paths on seven vertices Oliver Schaudt Universit at zu K

3-coloring graphs without induced paths on seven vertices Oliver Schaudt Universit at zu K oln, Institut f ur Informatik with Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Maya Stein, and Mingxian Zhong Graph coloring Graph coloring

598 views • 56 slides
DP-coloring of planar graphs Runrun Liu Central China Normal University May 17, 2019 Runrun Liu

DP-coloring of planar graphs Runrun Liu Central China Normal University May 17, 2019 Runrun Liu

DP-coloring of planar graphs Runrun Liu Central China Normal University May 17, 2019 Runrun Liu DP-coloring of planar graphs Definitions: proper coloring and list coloring A (proper) k-coloring of G is a mapping f : V ( G ) [ k ] such that

551 views • 37 slides
Implementing algorithms for path 3-coloring and path 3-list-coloring plane graphs Final Talk

Implementing algorithms for path 3-coloring and path 3-list-coloring plane graphs Final Talk

Implementing algorithms for path 3-coloring and path 3-list-coloring plane graphs Final Talk Aven Bross February 28, 2017 Committee Dr. Chappell (Chair) Dr. Lawlor Dr. Hartman Project goal Produce a nice, documented implementation

1.18k views • 73 slides
Graphs - Coloring, partition and structure Nicolas Bousquet Ecole Centrale Lyon 1/25

Graphs - Coloring, partition and structure Nicolas Bousquet Ecole Centrale Lyon 1/25

Graphs - Coloring, partition and structure Nicolas Bousquet Ecole Centrale Lyon 1/25 Representation of discrete structures 2/25 Graphs and coloring Definition A graph is a pair ( V , E ) where: V is a set of points called vertices.

1.37k views • 115 slides
Coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH

Coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH

Coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH Aachen with Flavia Bonomo, Maria Chudnovsky, Jan Goedgebeur, Peter Maceli, Maya Stein, and Mingxian Zhong Graph coloring Graph coloring a

921 views • 90 slides
Simple Graphs: timesoverlap. Coloring howmanygatesneeded? AlbertRMeyer, April5,2013

Simple Graphs: timesoverlap. Coloring howmanygatesneeded? AlbertRMeyer, April5,2013

Mathematics for Computer Science Flight Gates MIT 6.042J/18.062J flightsneedgates,but Simple Graphs: timesoverlap. Coloring howmanygatesneeded? AlbertRMeyer, April5,2013 AlbertRMeyer, April5,2013 coloring.1 coloring.2

351 views • 7 slides
Triangles and 3-coloring of planar graphs Bernard Lidick Iowa State University February 4,

Triangles and 3-coloring of planar graphs Bernard Lidick Iowa State University February 4,

Triangles and 3-coloring of planar graphs Bernard Lidick Iowa State University February 4, 2015 Outline what is graph coloring? Grtzschs theorem: triangle-free planar graphs are 3-colorable extensions of GT preserving

2.15k views • 106 slides
Coloring Count Cones of Planar Graphs Zden ek Dvo r ak Bernard Lidick y CanaDAM

Coloring Count Cones of Planar Graphs Zden ek Dvo r ak Bernard Lidick y CanaDAM

Coloring Count Cones of Planar Graphs Zden ek Dvo r ak Bernard Lidick y CanaDAM Vancouver, BC May 28, 2019 Motivation Theorem (4CT) Every planar graph is 4 -colorable. Problem Is there a polynomial-time algorithm to decide if a

1.04k views • 75 slides
Star-coloring planar graphs with high girth Daniel W. Cranston DIMACS, Rutgers

Star-coloring planar graphs with high girth Daniel W. Cranston DIMACS, Rutgers

Star-coloring planar graphs with high girth Daniel W. Cranston DIMACS, Rutgers dcransto@dimacs.rutgers.edu Joint with Craig Timmons and Andre Kundgen Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the

1.31k views • 83 slides
CS70 - Lecture 6 Graphs: Coloring; Special Graphs 1. Review of L5 2. Planar Five Color Theorem

CS70 - Lecture 6 Graphs: Coloring; Special Graphs 1. Review of L5 2. Planar Five Color Theorem

CS70 - Lecture 6 Graphs: Coloring; Special Graphs 1. Review of L5 2. Planar Five Color Theorem 3. Special Graphs: Trees: Three characterizations Hypercubes: Strongly connected! Administration You need to submit you grading option

514 views • 29 slides
Coloring Algorithms on Subcubic Graphs Harold N. Gabow, San Skulrattanakulchai

Coloring Algorithms on Subcubic Graphs Harold N. Gabow, San Skulrattanakulchai

Coloring Algorithms on Subcubic Graphs Harold N. Gabow, San Skulrattanakulchai hal@cs.colorado.edu, skulratt@cs.colorado.edu University of Colorado at Boulder Colorado, USA Coloring Algorithms on Subcubic Graphs p.1/56 Introduction Graph

1.2k views • 78 slides
Adjacent Vertex Distinguishing Definitions Some Known Results Colorings of Graphs Main Results

Adjacent Vertex Distinguishing Definitions Some Known Results Colorings of Graphs Main Results

Adjacent Vertex Distinguishing Definitions Some Known Results Colorings of Graphs Main Results Outline of Proofs Home Page Title Page Wang Weifan Page. 1 Total 80 Department of Mathematics Back To Full Screen

809 views • 80 slides
Chromatic Numbers Padal Nihar December 6, 2011 Padal Nihar Chromatic Numbers Introduction

Chromatic Numbers Padal Nihar December 6, 2011 Padal Nihar Chromatic Numbers Introduction

Introduction History Statement of Problem Dsatur Algorithm Applications References Chromatic Numbers Padal Nihar December 6, 2011 Padal Nihar Chromatic Numbers Introduction History Statement of Problem Dsatur Algorithm Applications

267 views • 12 slides
SLS at the Paul Scherrer Institute (PSI), Villigen, Switzerland SPring-8 12/03/02 Michael B

SLS at the Paul Scherrer Institute (PSI), Villigen, Switzerland SPring-8 12/03/02 Michael B

Top-up Operation at the Swiss Light Source SLS at the Paul Scherrer Institute (PSI), Villigen, Switzerland SPring-8 12/03/02 Michael B oge 1 Top-up Operation at the Swiss Light Source SLS Team at PSI SPring-8 12/03/02 Michael

488 views • 10 slides
Efficient methods for tune and chromaticity The NAFF method measurements in lepton and hadron

Efficient methods for tune and chromaticity The NAFF method measurements in lepton and hadron

Efficient methods for tune and chromaticity measurements in lepton and hadron rings P. Zisopoulos, Y. Papaphilippou Efficient methods for tune and chromaticity The NAFF method measurements in lepton and hadron rings Simulations Set-up

424 views • 23 slides
Capturing Realistic HDR Images Topics : Post-Processing. Sample Workflow. Q &

Capturing Realistic HDR Images Topics : Post-Processing. Sample Workflow. Q &

Capturing Realistic HDR Images (Part II) Dave Curtin Nassau County Camera Club April 4 th , 2016 Capturing Realistic HDR Images Topics : Post-Processing. Sample Workflow. Q & A. Capturing Realistic HDR Images

298 views • 28 slides
Valdez Operations Overview Alyeska Pipeline Service Company Prince William Sound Regional

Valdez Operations Overview Alyeska Pipeline Service Company Prince William Sound Regional

Valdez Operations Overview Alyeska Pipeline Service Company Prince William Sound Regional Citizens Advisory Council Board Meeting January 24, 2019 Anchorage, Alaska January 24, 2019 VMT Operations 2018* Tankers Loaded 247 Tankers Escorted

419 views • 23 slides
Welcome to the Music Department Maxwell Library, 3 rd Floor Room 312 Phone: 508-531-1377

Welcome to the Music Department Maxwell Library, 3 rd Floor Room 312 Phone: 508-531-1377

Welcome to the Music Department Maxwell Library, 3 rd Floor Room 312 Phone: 508-531-1377 General Information The Department of Music The Department of Music at Bridgewater State University provides a solid foundation in music history, theory,

561 views • 11 slides
JHR PROJECT. IRRADIATION DEVICES IN-SERVICE INSPECTION OF NUCLEAR PRESSURE EQUIPMENTS.

JHR PROJECT. IRRADIATION DEVICES IN-SERVICE INSPECTION OF NUCLEAR PRESSURE EQUIPMENTS.

JHR PROJECT. IRRADIATION DEVICES IN-SERVICE INSPECTION OF NUCLEAR PRESSURE EQUIPMENTS. Investigation of Non DESTRUCTIVE EXAMINATIONS For INSPECTION Purposes. IGORR 18 & IAEA Workshop S.Gaillot 03-07 th dec. 2017 CEA-France SYDNEY

193 views • 18 slides

More recommend