MS Project Defense The Study of Ramsey Numbers rCk, Ck,Ck

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MS Project Defense The Study of Ramsey Numbers
r(Ck, Ck,Ck)

• Yan Li
• January 21, 2004
• Committee
• Chairman Prof. Stanislaw P. Radziszowski
• Reader Prof. Peter G. Anderson
• Observer Prof. Ankur M. Teredesai

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Outline

• Motivation
• Mathematical background
• Literature
• Methods and Algorithms
• Results
• Conclusions
• Future work

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Why is Ramsey numbers?

• Party problem
• Classical problem in Ramsey theory
• Find the minimum number of guests that must be
  invited so that at least m will know each other
  or at least n will not know each other
• The solution is Ramsey number r(m, n)

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Ramsey numbers Graph theory

• Ramsey number is the minimum number of vertices v
  r(m, n) such that all undirected simple graphs
  on v vertices contain a clique of order m or an
  independent set of order n
• Clique Maximal complete sub-graph
• Independent set - A subset of the vertices such
  that no two vertices in the subset represent an
  edge of G

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Classical Ramsey numbers

• The classical Ramsey number r(r1, r2, , rm) is
  the smallest positive integer n such that for a
  complete graph Kn with any edge-coloring method
  (r1, r2, , rm), there always exists at least one
  monochromatic complete sub-graph Kri in color i

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Edge coloring

• Partitions of the edges of a complete graph
• Edge coloring (r1, r2, , rk), rigt0 for 0ltiltk1,
  is an assignment of one of k colors to each edges
  of a complete graph, such that it does not
  contain monochromatic complete sub-graph Kri in
  color i
• Adjacent edges may have the same color

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Ramsey numbers r(Ck, Ck, Ck)

• The smallest positive integer n such that any
  edge-coloring with three colors of the complete
  graph Kn must contain at least one monochromatic
  cycle of length k
• r(Ck, Ck, Ck) r3(Ck)

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Project goal

• The purpose of this project is to design and
  implement algorithms to verify the Ramsey number
  values of r3(Ck) and obtain the lower bounds on
  them
• G as undirected finite graph without any loops
  and multiple edges
• H as cycle Ck
• Algorithms dealing with multiple colors designed
  for specific case of 3 colors

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Isomorphism

• Let G and H be two graphs and f is a function
  from the vertex set of G to the vertex set of H.
  G and H are isomorphic if
• f is one-to-one
• f(v) is adjacent to f(w) in H iff v is adjacent
  to w in G

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Extremal graph number ext(H, n)

• The maximal number of edges of a graph with n
  vertices which does not contain a sub-graph
  isomorphic to H
• In this project, H considered as cycle Ck
• ext(C4, n) 1
• For example, ext(C4, 10) 16 1
• Also verified by my program

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Extremal graph

• The largest graph of order n which does not
  contain a sub-graph isomorphic to H
• Two extremal graphs of ext(C4, 10)
• Add any edge will generate C4
• Not isomorphic
• Firstly generated in 1 and verified by my
  program

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Good coloring (k n c)

• Three parameters of goodness of an edge coloring
  k n c
• k the number of vertices of a simple cycle
• n the number of vertices of a complete graph
• c the number of colors
• k-good c-colorings of Kn
• In this project, c is fixed to 3
• r3(Ck) is the smallest integer n for which there
  is no k-good 3-colorings of Kn

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Literature

• S. P. Radziszowski Small Ramsey Numbers
• http//www.combinatorics.org/Surveys
• Classical two color Ramsey numbers
• Classical multicolor Ramsey numbers
• r(3, 3, 3) 17 only known nontrivial value
• Multicolor special cases rm(Ck)
• Only five known exact values of r3(Ck)

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Brief history of r3(Ck)

• 1955 r2(3) 6 and r3(3) 17
• 1984 r3(4) 11
• Showed one 4-good 3-colorings of K10
• Proved the upper bound of r3(C4) be 11
• 1987 r3(4) 11
• 1992 r3(5) 17
• 1993 r3(6) 12
• 2003 r3(7) 25

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Approaches

• All methods dealing with graph instead of
  coloring
• Applied to check if a graph G contains any simple
  monochromatic path or cycle and construct Ck-free
  graphs
• Path search cycle search method
• Ck-free graph construction method

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Path search

• Checking if a graph G contains any simple
  monochromatic path Pk by eliminating one vertex
  and checking if there exists a path of length k-1
  recursively
• Input k G Output Boolean
• Steps
• Obtain the number of vertices of graph G
• Find the adjacent vertices of beginning vertex
  and store them as ai
• Eliminate the beginning vertex and the edges
  between the beginning vertex and its adjacent
  vertices, checks if there is a path k-1 between
  each ai and ending vertex. If yes, repeat till
  k1 otherwise, return 0

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Cycle search

• To find out if the edge coloring of Kn is k-good
  or not
• Input k G Output Boolean
• Checks if a graph G contains any simple
  monochromatic cycle Ck by checking if there is an
  edge between the beginning and ending vertices of
  the path based on the results of path searching
• Steps

A

• Check if there is an edge between vertices A and
  B
• If yes, check if there is a simple path Pk-1
  between A and B till k 1 if no, return 0

B
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Ck-free graph construction

• Construct all Ck-free graphs and obtain the
  extremal graph number
• Two cases
• Without isomorphism
• Including isomorphism
• Input k n Output a graph set containing all
  Ck-free graphs
• Steps
• Generates graphs on n vertices (without
  isomorphism or including isomorphism)
• Check cycle Ck and remove the graphs with cycle
• Obtain the extremal graph number

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Results

• Summary of number of graphs on 10 vertices
  without C4
• Only show graphs with edge number through 13 to
  16
• The total number of all graphs is 5069
• Proved ext(C4, 10) 16
• Proved ext(C6, 11) 23
• 3 extremal graphs of ext(C6, 11) exist

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Results (contd)

• Summary of number of graphs of ext(C4, 10)
• By eliminating isomorphism, the third column is
  identical with the second column
• Input data of matching algorithm

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Algorithms

• Two algorithms developed to verify the Ramsey
  number values r3(Ck)
• Direct extension algorithm (DE)
• Matching algorithm (MR)
• Trajectory algorithm applied to obtain the good
  lower bounds on r3(Ck)

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Direct extension

• An iterative point by point algorithm generating
  Ck-free
• 3-colorings of Kn by performing exhaustive
  enumerations
• Input n k Output set of k-good 3-colorings
  of Kn
• Step
• Construct three k-good 3-colorings of K3
• Add one vertex each time and perform exhaustive
  enumeration
• Remove the colorings with cycle Ck
• Successful with verifying Ramsey number values
  r3(C3) and r3(C4).
• However, large space complexity makes it not
  feasible for larger cycle

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1000 4-good 3-colorings of K10

• Summary of number of 4-good 3-colorings on K10 by
  DE
• number of vertices n number of all 4-good
  3-colorings
• 2 3
• 3 3
• 4 11
• 5 68
• 6 715
• 7 7580
• 8 38761
• 9 34009
• 10 1000
• 11 0

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1000 4-good 3-colorings of K10 (contd)

• No 4-good 3-colorings for K11 is a proof of
  r3(C4) 11
• Degree of each color less than or equal to 4
• The maximal number of edges is 16

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Verify the 1000 3-colorings are 4-good

• Isomorphism classes of each color from 1000
  3-colorings
• Graph sets S T
• S graphs with one color from 3-colorings of K10
• T 928 graphs on 10 vertices no C4 and e through
  13 to 16
• Many-to-one mapping between S and T
• Proof of all 1000 3-colorings are 4-good
• Only one graph in T appearing frequently in S

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Matching algorithm

• Based on the results of Ck-free graph
  construction
• Steps
• Find the extremal graph number ext(Ck, n)
• Obtain the possible color distribution cases
• Checks if there is an edge conflict between
  graphs with color 1 and color 2
• For those without edge conflict, check cycle Ck
  for color 3
• Overlap three graphs with color 1, 2 and 3 on Kn
• Verify r3(C4) 11

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Applying MR to r3(C4)

• Only three possible color distributions on edges
  of K10
• Overlap three graphs for each case

each entry of edges
each entry of graphs need to be overlapped
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Applying MR to r3(C4) (contd)

• Apply the graph set including isomorphism as
  color 2
• Steps
• All graphs in color 1 read in once and stored in
  memory
• Each time read in one graph of color 2 and check
  edge conflict
• Check cycle Ck of color 3
• Obtain 1000 4-good 3-colorings of k10
• None extend to 4-good 3-colorings of k11

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Trajectory algorithm

• A heuristic, not exhaustive algorithm to
  establish the lower bounds on Ramsey numbers
  r3(Ck)
• The idea is choosing a cycle-free 3-colorings of
  Kn randomly, generating more cycle-free
  3-colorings of Kn1 recursively until no
  cycle-free graphs, and repeating the trajectory
  generating as much as possible

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Results

• Verify the lower bounds of r3(C5), r3(C6) and
  r3(C7)
• Obtain the lower bounds on r3(C8) and r3(C10)
  first time

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Conclusions

• Obtain 1000 4-good 3-colorings of K10 by two
  independent algorithms DE and MR
• Verify Ramsey number values r3(C3) 17 and
  r3(C4) 11
• Verify the lower bounds on r3(C5), r3(C6) and
  r3(C7)
• Firstly obtain the lower bounds of r3(C8) and
  r3(C10)

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Software tools

• General utility program
• B. D.McKay http//cs.anu.edu.au/bdm/
• Specialized program
• C as development language
• Advantages of object-oriented features
• Ease of bit manipulation features
• Eight classes are designed

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Deliverables

• Project report
• Source code
• http//www.cis.rit.edu/yxl4059/ramsey.html

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Future work

• Improve the algorithms to verify the values of
  Ramsey numbers r3(C5), r3(C6) and r3(C7)
• Ramsey number with larger cycle r3(C9)

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References

• A. Bialostocki and J. Schonheim, On some Turan
  and Ramsey Numbers for C4, Graph Theory and
  Combinatorics, Academic Press,London, (1984)
  2933
• C. R. J. Clapham, The Ramsey Number r(C4, C4,
  C4), Periodica Mathematica, Vol. 18 (4), 1987,
  317-318
• R. Faudree, A. Schelten and I. Schiermeyer, The
  Ramsey Number r(C7, C7, C7), Discussions
  Mathematica Graph Theory, Vol. 23, 2003
• R. E. Greenwood and A. M. Gleason, Combinatorial
  Relations and Chromatic Graphs, Canadian Journal
  of Mathematics, 7 (1955), 1-7
• S. P. Radziszowski, Small Ramsey Numbers
• http//www.combinatorics.org/Surveys
• Y. Yang and P. Rowlinson, On the third Ramsey
  numbers of graphs with five edges, Journal of
  Combinatorial Mathematics and Combinatorial
  Computing, 11 (1992), 213-222
• Y. Yang and P. Rowlinson, On graphs without
  6-cycles and related Ramsey Numbers, Utilitas
  Mathematica, 44 (1993), 192-196

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The Study of Ramsey Number r(Ck, Ck,Ck)

• Thanks for your attention!
• Questions?