Measurements of the asymmetry of the emission peaks in the core of the Ca II H line for 105 giant stars are reported. The asymmetry is quantified with the parameter V/R, defined as the ratio between the maximum number of counts in the blueward peak and the redward peak of the emission profile. The Ca II H and K emission lines probe the differential motion of certain chromospheric layers in the stellar atmosphere. Data on V/R for the Ca II K line are drawn from previous papers and compared to

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... analogous H line ratio, the H and K spectra being from the same sets of observations. It is found that the H line V/R value is +0.04 larger, on average, than the equivalent K line ratio, however, the difference varies with B-V color. Red giants cooler than B-V = 1.2 are more likely to have the H line V/R larger than the K line V/R, whereas the opposite is true for giants hotter than B-V = 1.2. The differences between the Ca II H and K line asymmetries could be caused by the layers of chromospheric material from which these emission features arise moving with different velocities in an expanding outflow.