An ungraceful graph is a simple graph that does not possess a graceful labeling, i.e., a graph that is not a graceful graph (Gardner 1972). Such graphs have also been termed "disgraceful graphs" (Seoud and Wilson 1993).
The numbers of ungraceful graphs on 
, 2, ... nodes are 0, 1, 2, 4, 12, 30, 85, ... (OEIS A308556),
with the corresponding numbers of connected ungraceful graphs 0, 0, 0, 0, 3, 6, 34,
... (OEIS A308557), the first few of which
are illustrated above. As can be seen, there are no connected ungraceful graphs on
fewer than five vertives and exactly three connected ungraceful graphs on five vertices,
namely the butterfly graph (Horton 2003), cycle
graph 
,
and complete graph 
.
