Events where the two leading jets are separated by a pseudorapidity interval devoid of particle activity, known as jet-gap-jet events, are studied in proton-proton collisions at
√
s
=
13
 
 
TeV
. The signature is expected from hard color-singlet exchange. Each of the highest transverse momentum (
p
T
) jets must have
p
jet
T
>
40
 
 
GeV
and pseudorapidity
1.4
<
|
η
jet
|
<
4.7
, with
η
jet
1
η
jet
2
<
0
, where jet1 and jet2 are the leading and subleading jets in
p
T
, respectively. The analysis is based on data collected by the CMS and TOTEM experiments during a low luminosity, high-
β
∗
run at the CERN LHC in 2015, with an integrated luminosity of
0.66
 
 
pb
−
1
. Events with a low number of charged particles with
p
T
>
0.2
 
 
GeV
in the interval
|
η
|
<
1
between the jets are observed in excess of calculations that assume only color-exchange. The fraction of events produced via color-singlet exchange,
f
CSE
, is measured as a function of
p
jet
2
T
, the pseudorapidity difference between the two leading jets, and the azimuthal angular separation between the two leading jets. The fraction
f
CSE
has values of 0.4–1.0%. The results are compared with previous measurements and with predictions from perturbative quantum chromodynamics. In addition, the first study of jet-gap-jet events detected in association with an intact proton using a subsample of events with an integrated luminosity of
0.40
 
 
pb
−
1
is presented. The intact protons are detected with the Roman pot detectors of the TOTEM experiment. The
f
CSE
in this sample is
2.91
±
0.70
(
stat
)
+
1.08
−
1.01
(
syst
)
times larger than that for inclusive dijet production in dijets with similar kinematics.