Stephane G said:Prove by induction on the number of edges in a graph that any bipartite graph has edge colouring number equal to its maximum valency. Also, Find such an edge colouring for a bipartite 4-regular cartesian or tensor product of your choice of 2-regular graphs
Edge colouring of bipartite graphs is a graph theory problem that involves assigning colours to the edges of a bipartite graph in such a way that no two adjacent edges have the same colour.
A bipartite graph is a type of graph in which the vertices can be divided into two disjoint sets, such that all edges connect vertices from one set to vertices from the other set.
Proving valency equality is important because it ensures that the number of edges incident to each vertex in a bipartite graph is equal. This is necessary for a proper edge colouring, as each vertex must have an equal number of edges of each colour.
The valency of a vertex is the number of edges incident to that vertex. In a bipartite graph, the valency of each vertex should be equal to ensure a proper edge colouring.
There are several methods that can be used to prove valency equality in edge colouring of bipartite graphs, such as the degree sum formula, induction, and double counting.