A balanced edge coloring of a complete graph 
with 
colors is an assignment of colors to edges
in which the edges among any 
vertices use all 
colors (Xia and Szeider 2026). Incident edges
may receive the same color, and color classes need not be equally sized. Equivalently,
every color class, regarded as a spanning graph, has independence number at most 
.
For prime power 
, an affine plane of order
gives a balanced edge coloring of 
(Erdős and Gyárfás 1999). The Erdős-Gyárfás
balanced coloring conjecture concerns the largest complete
graph admitting such a coloring.
