---
tag: KNAAVNP65X
kind: statement
role: theorem
name: Galvin's theorem
sources:
  - wikipedia: en:920110@r1365511518
    locator: Dinitz theorem, section Galvin's theorem
    note: adopted
    mention: wikipedia://en/920110#T5PL
    sections: [Statement]
inventory: [J7FXTYFVZE]
---

## Statement

Galvin proved a considerably more general result: for every bipartite [[multigraph|0C62PZXX02]] $G$,

$$\chi'_\ell(G) = \chi'(G).$$

By [[Kőnig's edge-coloring theorem|0JP0ZF57PN]] the chromatic index of a bipartite multigraph equals its maximum degree, so $\chi'_\ell(G) = \Delta(G)$. Taking $G = K_{n,n}$ recovers the Dinitz theorem.