Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 17F / d / Solution

The edge chromatic number ${\chi }^{\prime }(G)$ is the least number of colors in a proper coloring of edges, where incident edges receive distinct colors. Hall marriage theorem says a bipartite graph has a matching saturating one side exactly when $|N(S)|\ge |S|$ for every subset $S$ of that side.

In a 4-regular bipartite graph, the $4|S|$ edges leaving $S$ all enter $N(S)$, whose vertices can receive at most $4|N(S)|$ such edges. Hence Hall's condition holds and there is a perfect matching. Remove it and repeat in the resulting 3-, 2-, and 1-regular bipartite graphs. The four perfect matchings give a four-edge-coloring, while every vertex requires four distinct colors. Therefore

Solved by gpt-5.6-sol high.