Bipartite graphs,Complete bipartite graph, Four color theorem, Applications of Graph Coloring

1. Topic
1. Bipartite graphs coloring.
2. Four color theorem.
3. Applications of Graph Coloring.

2. Bipartite Graph
1. A graph G is bipartite if the node set V can be
partitioned into two sets V1 and V2 in such a way
that no nodes from the same set are adjacent.
2. The sets V1 and V2 are called the color classes of
G and (V1, V2) is a bipartition of G. In fact, a graph
being bipartite means that the nodes of G can be
colored with at most two colors, so that no two
adjacent nodes have the same color.

3. Bipartite Graph
3. We will depict bipartite graphs with their nodes colored
black and white to show one possible bipartition.
4. We will call a graph m by n bipartite if |V1| = m and |V2| = n
and a graph is balanced bipartite graph when |V1| = |V2|.

4. Bipartite graphs coloring
Suppose there are two colors: blue and red. Color the first
vertex blue. For each newly-discovered node, color it the
opposite of the parent (i.e. red if parent is blue). If the child
node has already been discovered, check if the colors are the
same as the parent. If so, then the graph isn’t bipartite. If the
traversal completes without any conflicting colors, then the
graph is bipartite.

5. Four color theorem
The four color theorem, states that, given any separation of a
plane into contiguous regions, producing a figure, no more
than four colors are required to color the regions of the figure
so that no two adjacent regions have the same
color. Adjacent means that two regions share a common
boundary curve segment, not merely a corner where three or
more regions meet.

6. Applications of Graph Coloring
• Making schedule or time table of exam.
• Mobile radio frequency assignment.
• Register Allocation
• Map Coloring and Bipartite Graphs
• Sudoku is also a variation of Graph coloring.

7. That’s all
Thank you