Graph Coloring and Hamiltonian cycles

1
Graph Coloringand Hamiltonian cycles

• Lecture 22
• CS 312

2
Follow up

• 8 queens material.

3
Objectives

• Use a backtracking algorithm to solve graph
  coloring.
• Solve the Hamiltonian cycle problem
• Discuss differences between DFS and BFS
  backtracking algorithms.

4
Graph Coloring Problem

• Assign colors to the nodes of a graph so that no
  adjacent nodes share the same color
• nodes are adjacent if there is an edge for node i
  to node j.
• Find all m-colorings of a graph
• all ways to color a graph with at most m colors.

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Graph coloring algorithm
mColor (thisNode) while (true) nextColoring
(thisNode) if (colorthisNode 0) then
break // no more colors for thisNode if
(thisNode numNodes) then print this
coloring // found a valid coloring of all nodes.
else mColor (thisNode 1) // try to color
the next node. endWhile nextColoring
(thisNode) while (true) colorthisNode
(colorthisNode 1) mod (numColors 1) if
(colorthisNode 0) then return // no more
colors to try. for k 1 to numNodes1 if
(connectedk,thisNode and colork
colorthisNode) then break endfor if
(k numNodes1) return // found a new color
because no nodes clashed. endWhile
6
M-coloring function

• void mColoring(int k)
• do //Generate all legal assignments for xk
• NextValue(k)
• if (!xk) break//No new color possible
• if (kn)//At most m colors have been used to
  color the n vertices.
• for (int i1iltni) coutltltxiltlt
• cout ltlt endl
• else mColoring (k1)
• while(1)
• void NextValue(int k)
• do
• xk (xk1)(m1)//next highest color
• if (!xk) return //All colors have been used.
• for (int j1jltnj) //Check if this color
  is distinct
• if (Gkj //If (k, j) is an edge
• (xk xj)) //and if adj. vertices
• break //have the same color
• if (j n1) return //New color found

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Small Example
8
Aside Coloring a Map

• Assign colors to countries so that no two
  countries are the same color.
• Graph coloring as map coloring.

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Map coloring as Graph Coloring
Idaho
Wyoming
Utah
Nevada
Colorado
New Mexico
Arizona
10
Four color theorem.

• How many colors do you need?
• Four.
• Haken and Appel using a computer program and
  1,200 hours of run time in 1976. Checked 1,476
  graphs.
• First proposed in 1852.

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Hamiltonian Cycles

• Given a graph with N vertices.
• Find a cycle that visits all N vertices exactly
  once and ends where it started.
• Sound familiar?

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Example
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Hamiltonian
Hamiltonian (k) while (1) xk
NextValue(k) if (xk 0) then return if
(k N) then print solution else
Hamiltonian (k1) endWhile
NextValue (k) while (1) value (xk1) mod
(N1) if (value 0) then return value if
(Gxk-1,value) for j 1 to k-1 if
xj value then break if (jk) and (k lt
N or k N and GxN,x1) then
return value endWhile
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Hamiltonian functions

• void Hamiltonian(int k)
• do //Generate values for xk
• NextValue(k) //Assign a legal next value to
  xk
• if (!xk) return
• if (kn)
• for (int i1iltni) coutltltxiltlt
• cout ltlt 1\n
• else Hamiltonian(k1)
• while(1)
• void NextValue(int k)
• do
• xk (xk1)(n1)//next vertex
• if (!xk) return
• if (Gxk-1xk) //Is there an edge?
• for (int j1jltk-1j) if (xjxk) break
  if (jk) //if true, then the vertex is
  distinct.
• if ((kltn) (kn) Gxnx1))
• return
• while (1)

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Thats it.

• Have a good weekend.
• Midterm 2 is next week.