The odd Hadwiger conjecture, proposed by Gerards and Seymour (Jensen and Toft 1995), is a strengthening of the Hadwiger conjecture.
It states that, if a graph 
has chromatic number 
, then 
has the complete graph 
as an odd minor.
Here, an odd 
minor consists of 
vertex-disjoint trees 
, 
, 
in 
, each properly 2-colored, such that every pair of trees is
joined by an edge whose endpoints have the same color.
The conjecture is true for 
and 
, and the case 
is a strengthening of the four-color
theorem (Kühn et al. 2025). However, the conjecture was disproved
by Kühn et al. (2025), who constructed graphs with no odd 
minor and chromatic
number at least 
for sufficiently large 
. The disproof of the odd Hadwiger conjecture does not disprove
the Hadwiger conjecture itself, which remains
open for larger values of 
.
