Let k ≥ 1, and let G be a k-regular bipartite graph with bipartition X, Y . Prove that |X| is the minimum size of a vertex cover in G.

Let G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible number of edges in G? Show that this bound is sharp.

Let G be a simple graph with n vertices and k connected components. Find the maximum and the minimum possible number of edges of G

Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex. Let G be a graph of order n and size strictly less than n - - 1. Prove that G is not connected.

Let G be a connected graph that has exactly 4 vertices of odd degree: v1, V2, V3 and V4. Show that there are paths with no repeated edges from v1 to V2, and from v3 to v4, such that every edge in G is in exactly one of these paths.

A graph is bipartite if its vertex set can be partitioned into two sets V₁ and V2 such all edges are between V₁ and V2 (i.e. there are no edges joining vertices inside V₁, and the same for V2). (a) Draw a bipartite graph with 5 vertices and 5 edges. (b) What is the maximum number of edges for a bipartite graph with 2n vertices (suppose n > 1)?

Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and let m be the minimum degree of the vertices of G. Show that  2e/v≥m

Prove that for any two edges of a 2-connected graph, a cycle exists containing both of them.

Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7 vertices of degree 8 or at least 6 vertices of degree 7. Do not use the planar equation e <= 3v - 6.

Solve 5b