The formation and properties of star clusters formed at the edges of H II regions are poorly known. We study stellar content, physical conditions, and star formation processes around a relatively unknown young H II region IRAS 10427-6032, located in the southern outskirts of the Carina Nebula. We make use of near-IR data from VISTA, mid-IR from Spitzer and WISE, far-IR from Herschel, sub-mm from ATLASGAL, and 843 MHz radio-continuum data. Using multi-band photometry, we find a total of 5 Class I and 29 Class II young stellar object (YSO) candidates, most of which newly identified, in the 5$times$5 region centered on the IRAS source position. Modeling of the spectral energy distribution for selected YSO candidates using radiative transfer models shows that most of these candidates are intermediate mass YSOs in their early evolutionary stages. A majority of the YSO candidates are found to be coincident with the cold dense clump at the western rim of the H II region. Lyman continuum luminosity calculation using radio emission indicates the spectral type of the ionizing source to be earlier than B0.5-B1. We identified a candidate massive star possibly responsible for the H II region with an estimated spectral type B0-B0.5. The temperature and column density maps of the region constructed by performing pixel-wise modified blackbody fits to the thermal dust emission using the far-IR data show a high column density shell-like morphology around the H II region, and low column density (0.6 $times$ 10$^{22}$ cm$^{-2}$) and high temperature ($sim$21 K) matter within the H II region. Based on the morphology of the region in the ionized and the molecular gas, and the comparison between the estimated timescales of the H II region and the YSO candidates in the clump, we argue that the enhanced star-formation at the western rim of the H II region is likely due to compression by the ionized gas.
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