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 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 5-cycle
exists in 
(S. Wagon, pers. comm., Feb. 2013).
Since a chordless cycle is a vertex-induced subgraph that is also a regular graph of vertex degree 2, the Haemers
regular upper bound shows that if 
is a regular graph of vertex degree 
on 
vertices with least graph
eigenvalue 
,
then every chordless cycle in 
has length at most
 |
(Haemers 1995, Thm. 3.5), where 
is the floor function.
For strongly regular graphs, the regular adjacency bounds can give a smaller upper bound by using the full parameter set.
No chordless cycles (of length four or more) of length greater than 
exist in a graph 
with independence number 
.
No chordless cycles of length greater than 
exist in the graph
complement 
,
where 
is the clique number of 
. This follows from the preceding bound since 
.
Every cycle of a cactus graph has no cycle chord, but there exist graphs (e.g., the theta0-graph and Pasch graph) whose cycles all have no cycle chord but which are not cactus graphs.
