The Hadwiger number of a graph 
, variously denoted 
(Zelinka 1976, Ivančo 1988) or 
(Stiebitz 1990), is the number of vertices in the largest
complete minor of
(Hadwiger 1943, Fowler et al. 2022).
The Hadwiger number is also called the contraction clique number (Bollobás
et al. 1980) or homomorphism degree (Halin 1976).
The Hadwiger conjecture states that for any loopless graph 
,
 |
(1)
|
where 
the chromatic number of 
(Hadwiger 1943).
Forests have 
, graphs with treewidth
at most 2 have 
,
planar graphs have 
, and apex graphs have
.
The graphs with Hadwiger number at most five include the apex graphs and the linklessly embeddable
graphs (Robertson et al. 1993), both of which have the complete
graph 
among their forbidden minors.
Computing the Hadwiger number of a graph is an NP-hard problem.
For a graph 
and its graph complement 
, the Hadwiger number satisfies
 |
(2)
|
(Kostochka 1984, Stiebitz 1990).
The Hadwiger number of a finite simple connected graph with vertex cuts is equal to the maximum of the Hadwiger numbers of its blocks, and the Hadwiger number of a finite simple disconnected graph is equal to the maximum of the Hadwiger numbers of its connected components (Zelinka 1976).
The Hadwiger number of the complete bipartite graph 
is
 |
(3)
|
(Zelinka 1976), of the complete 
-partite graph 
is
 |
(4)
|
for 
and 
(Ivančo 1988), and of the wheel complement
graph 
is
 |
(5)
|
for 
,
where 
is the floor function (Fowler et al. 2022).
