1. Graph Coloring and 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.

5. Graph coloring algorithm mColor (thisNode) while (true) nextColoring (thisNode) if (color[thisNode] == 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) color[thisNode] = (color[thisNode] + 1) mod (numColors + 1) if (color[thisNode] == 0) then return // no more colors to try. for k = 1 to numNodes+1 if (connected[k,thisNode] and color[k] == color[thisNode]) then break endfor if (k == numNodes+1) return // found a new color because no nodes clashed. endWhile

6. M-coloring function void mColoring(int k) {do {//Generate all legal assignments for x[k] NextValue(k); if (!x[k]) break;//No new color possible if (k==n){// At most m colors have been used to color the n vertices. for (int i=1;i<=n;i++) cout<<x[i]<<‘ ‘; cout << endl; else mColoring (k+1); }while(1);} void NextValue(int k) {do { x[k] = (x[k]+1)%(m+1);//next highest color if (!x[k]) return; //All colors have been used. for (int j=1;j<=n;j++) { //Check if this color is distinct if (G[k][j] //If (k, j) is an edge && (x[k] == x[j])) //and if adj. vertices break; //have the same color } if (j == n+1) return; //New color found }while (1);//Otherwise try to find another color.}

7. 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.

9. Map coloring as Graph Coloring UtahNevada Idaho Colorado New Mexico Wyoming 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.

11. 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?

12. Example

13. Hamiltonian Hamiltonian (k) while (1) x[k] = NextValue(k) if (x[k] = 0) then return if (k == N) then print solution else Hamiltonian (k+1) endWhile NextValue (k) while (1) value = (x[k]+1) mod (N+1) if (value == 0) then return value if (G[x[k-1],value]) for j = 1 to k-1 if x[j] = value then break if (j==k) and (k < N or k == N and G[x[N],x[1]]) then return value endWhile

14. Hamiltonian functions void Hamiltonian(int k) {do {//Generate values for x[k] NextValue(k); //Assign a legal next value to x[k] if (!x[k]) return; if (k==n) { for (int i=1;i<=n;i++) cout<<x[i]<<‘ ‘; cout << “1\n”; } else Hamiltonian(k+1); }while(1);} void NextValue(int k) {do { x[k] = (x[k]+1)%(n+1);//next vertex if (!x[k]) return; if (G[x[k-1][x[k]) { //Is there an edge? for (int j=1;j<=k-1;j++) if (x[j]==x[k]) break; if (j==k) //if true, then the vertex is distinct. if ((k<n) || (k==n) && G[x[n]][x[1]])) return; } }while (1); }

15. That’s it. Have a good weekend. Midterm 2 is next week.