Problem: Prove that a $d$-regular bipartite graph is $d$-edge colourable using Konig-Egervary theorem
If the the graph $G$ is bipartite then the size of max-matching equals to the size of min-vertex-cover
Here is my proof for the problem: Create a Matching: Apply Kőnig's theorem to find a maximum matching M in G. Since G is d-regular, each vertex in U and V is incident to exactly d edges in the matching.
Vertex Cover: By Kőnig's theorem, the size of the maximum matching is equal to the size of the minimum vertex cover. Let C be a minimum vertex cover in G.
Coloring: Now, we can use the vertices in U - C as one color class and the vertices in V ∩ C as the other color class. Since each vertex in U is adjacent to exactly one vertex in V in the matching M, and vice versa, the resulting coloring is proper.
Specifically, each vertex in U - C has all its neighbors in V ∩ C, and each vertex in V ∩ C has all its neighbors in U - C. Therefore, the coloring is proper, and each vertex is assigned one of the d colors.
I hope that you can help me to check and indicate if there is anything wrong