Question

An undirected bipartite graph has n vertices and m edges.

a) If the graph is connected, what is the minimum number of edges?

b) If the graph is disconnected, what is the maximum number of edges?

c) What is the longest single path?

d) If the path can pass through a vertex and not any edges more than once, What is the longest path?

Kindly provide me with an example for me to relate

Answer 1

(b) If the graph is disconnected then the number of edges will be zero. Because of two partites are different, there is no edge link between them

(a) If the graph is connected then the minimum number of edges is one because at worst case the only vertex of one partite is connected to one vertex of other partite.

(c) Let the bipartite graph is over (V1,V2) and we need to find the longest possible augmenting path for the graph. If so, the longest path is an alternating one like s→v1→u1→⋯→vℓ→uℓ→t where vi's are in V1 and ui's are in V2 and ℓ is at most min{|V1|,|V2|}. Thus, the maximum possible length is 2*min{|V1|,|V2|}+1.