  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 $\sqrt{s} =$ 13 TeV. The signature is expected from
hard color-singlet exchange. Each of the highest transverse momentum
($p_\mathrm{T}$) jets must have $p_\mathrm{T}^\text{jet}$ $\gt$ 40 GeV and
pseudorapidity 1.4 $\lt$ $|\eta^\text{jet}|$ $\lt$ 4.7, with $\eta^\text{jet1}
\eta^\text{jet2}$ $\lt$ 0, where $\text{jet1}$ and $\text{jet2}$ are the
leading and subleading jets in $p_\mathrm{T}$, respectively. The analysis is
based on data collected by the CMS and TOTEM experiments during a low
luminosity, high-$\beta^*$ 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 $p_\mathrm{T}$ $\gt$ 0.2 GeV in the interval $|\eta|$ $\lt$ 1 between the
jets are observed in excess of calculations that assume only color-exchange.
The fraction of events produced via color-singlet exchange, $f_\text{CSE}$, is
measured as a function of $p_\mathrm{T}^\text{jet2}$, the pseudorapidity
difference between the two leading jets, and the azimuthal angular separation
between the two leading jets. The fraction $f_\text{CSE}$ 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 $f_\text{CSE}$ in this sample is 2.91 $\pm$ 0.70 (stat)
$^{+1.08}_{-1.01}$ (syst) times larger than that for inclusive dijet production
in dijets with similar kinematics.
