We have taken 2.2 mum window spectra of the nuclear regions of a sample of 19 luminous infrared galaxies and used these to determine the 2.3 mum CO absorption feature. We have found a relationship between the depth of the 2.3 mum absorption bands and the H - K color of the galaxies. Galaxies with H - K colors in the range 0.4-0.8 tend to have deeper CO absorption than normal galaxies. Their near-infrared emission is therefore probably dominated by late-type supergiant or metal-rich giant stars that have been reddened by interstellar dust. Galaxies with H - K colors redder than 0.9 tend to have much weaker than normal CO absorption. Their near-infrared emission therefore cannot be simply reddened starlight from late-type giant stars, and they are likely to contain an additional source of near-infrared radiation. The most likely possibilities are an active galactic nucleus (AGN), or emission from very small transiently heated dust grains. Of the five objects with reddest H - K colors and weakest CO absorption, three (IRAS 05189 - 2524, UCG 5 101 and Mrk 231) belong to the ultraluminous class of IRAS galaxies and are known to have Seyfert nuclei; two are off-nuclear components of the merging system Arp 299. We have also detected for the first time strong emission lines of molecular hydrogen in the infrared luminous galaxy NGC 4418, the galaxy with the deepest known 10 mum silicate absorption feature.