Bipartite Graph

Description:

• A simple graph
• Its vertex set 𝑉 can be partitioned into two disjoint sets $V_1$ and $V_2$ such that every edge in the graph connects a vertex in $V_1$ and a vertex in $V_2$
• We call the pair $(V_1,V_2)$ a bipartition of the vertex set 𝑉 of 𝐺.
• Some graphs that are bipartite:
  • Cycle graphs with even number of vertices
  • Every tree graph
  • Hypercube
• Chromatic Number of bipartite graph is 2
  • A graph, $G$, with at least one edge is bipartite if and only if $\chi (G)=2$theorem
• Tree is also a bipartite
• $K_{5,3}$: pic

Theorem:

• theorem 𝐺 is bipartite if and only if 𝐺 is connected and has no odd-length cycles (cycle with odd number of vertices).

Complete bipartite graph:

• All vertices in $V_1$ is connected to all vertices in $V_2$.

Matching in a bipartite graph