The strong Roberson conjecture states that a class 
of finite simple graphs
closed under graph minors and disjoint
union is homomorphism distinguishing closed (Roberson 2026). Explicitly, for
every graph 
,
there should exist graphs 
such that
 |
but 
, where 
counts graph homomorphisms
from 
to 
.
Kristjánsson (2026) disproved the conjecture. For the circulant graph 
, counts from all graphs
excluding 
as a graph minor determine the count from 
itself. The corresponding class
is closed under graph minors and disjoint
union, but adding 
does not increase its distinguishing power. Replacing every
graph vertex of 
by two vertices joined by
a graph edge, with the two incoming and two outgoing
edges assigned to opposite ends according to the positive
jumps of the circulant graph, gives a cubic
graph on 182 vertices. This bipartite
graph supplies a counterexample both for graph minors
and for the corresponding conjecture with graph immersions in their place. The weak Roberson conjecture remains open. Kristjánsson
(2026) reports substantial AI collaboration in developing these mathematical results.
