I. INTRODUCTION The present knowledge of quantum chromodynamics (QCD) suggests that quark matter might be in di ff erent color superconducting phases at high densities. Thus, one expects the core of the hybrid stars to be composed of color superconducting quark matter (CSQM) surrounded by a nuclear mantle. The possible CSQM phases are the two-flavor c olor superconductor (2SC) [1–3], the colour flavour locked (CFL) phase [4, 5], and crystallinecolor superconductor (CCS) [6, 7]. The speculation that the CSQM exists in the core of the hybrid stars has triggered many theoretical investigations both on the modeling of the equation of state (EOS) of quark matter and on the phenomenological signatures of the presence of quark matter in the compact stars [8]. The nuclear matter phase of the hybrid star is described by the various models which can be broadly grouped into (i) non-relativistic potential models [9], (ii) non-relativistic mean-field models [10–13], (iii) field theoretical based relativistic mean-field models [14–16] and (iv) Dirac- Brueckner-Hartree-Fock model [17–20]. The quark matter inthe colour superconducting phases are usually described either within the MIT bag model or using a more realistic NJL-like model. The studies based on the MIT bag model indicate the existence of stable configurations of hybrid stars with the CFL quark matter core [21–24]. Further, the MI T bag model predicts the absence of the 2SC colour superconducting phase in the hybrid stars [25]. The scenario is some what di ff erent when NJL model is employed to study the hybrid stars with CSQM core. The stable configurations of hybrid stars with 2SC quark matter core arepossible within the NJL model [26– 29]. However, earlier investigations [30–32] based on the N JL model ruled out the possibility of CFL quark matter at the core of the hybrid stars, because, it rendered the hybrid star unstable. Only very recently [33, 34], it has been demonstrated that inclusion of the six-fermion interaction term together with large enough values of the scalar diquark coupling strength in the NJL model can yield stable configurations of the hybrid star containin g 2SC or CFL quark matter core. The NJL model is also applied to study the possibility of existence of the CCS quark matter phase in the hybrid stars [35, 36]. The stability and the structure of the non-rotating hybrid stars are quite sensitive to the choice of the EOS of the nuclear matter and the quark matter [21, 37]. Further, one often finds that even though the stable configurations of the non-rotating hybridstar for a given EOS belong to the third family of compact stars, but, the maximum rotation frequency upto which these hybrid stars are stable is much lower than the corresponding mass-shedding (Keplerian) frequency [38, 39]. 2