In 1943, Hadwiger conjectured that every graph with no
K
t
K_t
minor is
(
t
−
1
)
(t-1)
-colorable for every
t
≥
1
t\ge 1
. In the 1980s, Kostochka and Thomason independently proved that every graph with no
K
t
K_t
minor has average degree
O
(
t
log
t
)
O(t\sqrt {\log t})
and hence is
O
(
t
log
t
)
O(t\sqrt {\log t})
-colorable. Recently, Norin, Song and the second author showed that every graph with no
K
t
K_t
minor is
O
(
t
(
log
t
)
β
)
O(t(\log t)^{\beta })
-colorable for every
β
>
1
/
4
\beta > 1/4
, making the first improvement on the order of magnitude of the
O
(
t
log
t
)
O(t\sqrt {\log t})
bound. The first main result of this paper is that every graph with no
K
t
K_t
minor is
O
(
t
log
log
t
)
O(t\log \log t)
-colorable.
This is actually a corollary of our main technical result that the chromatic number of a
K
t
K_t
-minor-free graph is bounded by
O
(
t
(
1
+
f
(
G
,
t
)
)
)
O(t(1+f(G,t)))
where
f
(
G
,
t
)
f(G,t)
is the maximum of
χ
(
H
)
a
\frac {\chi (H)}{a}
over all
a
≥
t
log
t
a\ge \frac {t}{\sqrt {\log t}}
and
K
a
K_a
-minor-free subgraphs
H
H
of
G
G
that are small (i.e.
O
(
a
log
4
a
)
O(a\log ^4 a)
vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small
K
t
K_t
-minor-free graphs, we show that
K
t
K_t
-minor-free graphs are
O
(
t
log
log
t
)
O(t\log \log t)
-colorable. Second, it shows that proving Linear Hadwiger’s Conjecture (that
K
t
K_t
-minor-free graphs are
O
(
t
)
O(t)
-colorable) reduces to proving it for small graphs. Third, we prove that
K
t
K_t
-minor-free graphs with clique number at most
log
t
/
(
log
log
t
)
2
\sqrt {\log t}/ (\log \log t)^2
are
O
(
t
)
O(t)
-colorable. This implies our final corollary that Linear Hadwiger’s Conjecture holds for
K
r
K_r
-free graphs for every fixed
r
r
; more generally, we show there exists
C
≥
1
C\ge 1
such that for every
r
≥
1
r\ge 1
, there exists
t
r
t_r
such that for all
t
≥
t
r
t\ge t_r
, every
K
r
K_r
-free
K
t
K_t
-minor-free graph is
C
t
Ct
-colorable.
One key to proving the main theorem is building the minor in two new ways according to whether the chromatic number ‘separates’ in the graph: sequentially if the graph is the chromatic-inseparable and recursively if the graph is chromatic-separable. The other key is a new standalone result that every
K
t
K_t
-minor-free graph of average degree
d
=
Ω
(
t
)
d=\Omega (t)
has a subgraph on
O
(
t
log
3
t
)
O(t \log ^3 t)
vertices with average degree
Ω
(
d
)
\Omega (d)
.