Enumerative chromatic choosability is the property of a graph 
that its list color function equals its chromatic polynomial for every positive
integer 
:
 |
Since both sides vanish below the chromatic number 
,
it is equivalent to require the equality only for 
. It implies chromatic choosability, the equality
of choice number and chromatic
number, but the converse fails (Allred and Mudrock 2026).
A connected graph with chromatic number 2 is enumeratively chromatic-choosable iff its
core, obtained by successively deleting vertices
of vertex degree 1, is 
, an even cycle graph, or
the complete bipartite graph 
(Allred and Mudrock 2026, Theorem 10). For disconnected
graphs, the property holds if it holds for every connected
component, since both counting functions multiply over components. The converse
fails. For example, the graph disjoint union 
is enumeratively chromatic-choosable although its theta
graph component is not. For 
, both counting functions of the union vanish because
of 
,
while for 
the two functions agree on both components (Allred and
Mudrock 2026, Theorem 11).
The theta graph 
, consisting of three internally disjoint paths
of the indicated lengths joining the same two vertices,
is chromatic-choosable but fails enumerative chromatic choosability for 
. Nevertheless, its two counting functions agree for
every 
, and its graph join with
is enumeratively chromatic-choosable (Allred and Mudrock 2026, Theorems 11 and
29). Allred and Mudrock (2026, Conjecture 6) conjecture that every graph 
has some complete graph 
for which 
is enumeratively chromatic-choosable.
