A chordless cycle of a graph 
is a graph cycle in 
that has no cycle chord. Unfortunately,
there are conflicting conventions on whether or not 3-cycles should be considered
chordless. In particular, in mathematical graph theory, "trivial" cycles
of length 3 are commonly not considered chordless (e.g., West 2000), while
in computer science, length-3 cycles are generally considered chordless (e.g.,
Cook et al. 2013, Wikipedia 2020). For example, (West 2000, p. 225) states,
"A chordless cycle in 
is a cycle of length at least 4 in 
that has no chord (that is, the cycle is an induced subgraph),
while Cook et al. (2013, p. 197) states, "a triangle is considered to
be a chordless cycle."
Excluding 3-cycles allows for simpler definitions and theorem statements (particularly those related to perfect graphs), for example permitting the definition of chordal graph as a simple graph possessing no chordless cycles (West 2000, p. 225) without further qualification.
The "ChordlessCycles" and related properties in the Wolfram Language function GraphData adopt the convention of West (2000, p. 225) that chordless cycles must have length at least 4.
An alternate approach followed by Chvátal defines a graph hole as "a chordless cycle of length at least four," thus distinguishing between the a generic "chordless cycle" (possibly allowing length-3 cycles) and a "hole" (excluding them).
Since the term "chordless cycle" seems to be used much more widely than "graph hole," perhaps the clearest approach is to always state "chordless cycle of length at least four" when length-3 cycles are to be excluded.
The numbers of chordless cycles of each possible length may be encoded in a polynomial here termed the chordless cycle polynomial.
A graph is perfect iff neither the graph nor its complement has a (length four or greater) odd chordless cycle.
If a chordless 5-cycle exists in a graph 
, one also exists in its graph
complement 
since in the complement the interior diagonals are really edges in the original.
In addition, if no 5-cycle exists in 
, then no chordless cycle exists in 
(S. Wagon,. pers. comm., Feb. 2013).
No chordless cycles (of length four or more) of length greater than 
exist in a graph 
with independence number 
.
No chordless cycles (of length four or more) of length greater than 
exist in the graph
complement 
of a graph 
with 
,
where 
is the clique covering number and 
is the clique number.
Every cycle of a cactus graph is chordless, but there exist graphs (e.g., the 
-graph and Pasch graph)
whose cycles are all chordless but which are not cactus
graphs.
