Let G be a bipartite graph partitioned into vertex sets V and W. Assume all vertices have the same degree.

Show G has a perfect matching.

Relationship between feasible potentials and negative dicycles.

Let G be a directed graph where c_e is the cost of arc e. If the nodes of G can be assigned feasible potentials then G has no negative dicycle.

Discrete. Send response as attachment

Please see the attached file for full problem description.

Knapsack Problem (Write in Canonical Form)

Let [EQUATION1] with [EQUATION2] and [EQUATION3]. The idea is to write each such set in some simple canonical form. (i) When n = 2, how many distinct knapsack sets are there? Write them out in a canonical form with integral coefficients and 1 = [EQUATION4]. (ii) Repeat for n = 3 with [EQUATION5]. *(For proper equations and complete problem, please see the attachment)

Discrete Math - Set Theory / Identity / Proof

Discrete Math - Set Theory / Identity / Proof For all sets A, B, and C, Prove the "attachment" with the use of laws such as de morgan, distributive, etc. Need a sample step by step in solving the problem