Abstract
Let
P
G
(q)
denote the chromatic polynomial of a graph
G
on
n
vertices. The `shameful conjecture' due to Bartels and Welsh states that,
P
G
(n)
P
G
(n−1)
≥
n
n
(n−1
)
n
.
Let
μ(G)
denote the expected number of colors used in a uniformly random proper
n
-coloring of
G
. The above inequality can be interpreted as saying that
μ(G)≥μ(
O
n
)
, where
O
n
is the empty graph on
n
nodes. This conjecture was proved by F. M. Dong, who in fact showed that,
P
G
(q)
P
G
(q−1)
≥
q
n
(q−1
)
n
for all
q≥n
. There are examples showing that this inequality is not true for all
q≥2
. In this paper, we show that the above inequality holds for all
q≥36
D
3/2
, where
D
is the largest degree of
G
. It is also shown that the above inequality holds true for all
q≥2
when
G
is a claw-free graph.