An odd cycle decomposition of fixed length 
is a graph decomposition
into cycle graphs 
, where 
is odd. Necessary divisibility conditions are that every
vertex degree is even and that 
divides the edge count. A graph satisfying these conditions is called 
-divisible (Hou et al. 2026).
Hou et al. (2026) proved that, for every fixed odd 
and every 
, all sufficiently large 
-vertex 
-divisible simple graphs 
with
 |
have such a decomposition, where 
is the minimum
vertex degree. The coefficient 
is best possible. Their new cases are 
, 7, ..., 71, completing the thresholds for all fixed odd
lengths. For 
,
this coefficient is 
. Hou et al. (2026) acknowledge generative AI assistance
with proof development and checking, and take responsibility for the mathematical
content.
