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We derived galaxy colour selections from Euclid and ground-based photometry,aiming to accurately define background galaxy samples in cluster weak-lensinganalyses. Given any set of photometric bands, we developed a method for thecalibration of optimal galaxy colour selections that maximises the selectioncompleteness, given a threshold on purity. We calibrated galaxy selectionsusing simulated ground-based $griz$ and Euclid $Y_{\rm E}J_{\rm E}H_{\rm E}$photometry. Both selections produce a purity higher than 97%. The $griz$selection completeness ranges from 30% to 84% in the lens redshift range$z_{\rm l}\in[0.2,0.8]$. With the full $grizY_{\rm E}J_{\rm E}H_{\rm E}$selection, the completeness improves by up to $25$ percentage points, and the$z_{\rm l}$ range extends up to $z_{\rm l}=1.5$. The calibrated colourselections are stable to changes in the sample limiting magnitudes andredshift, and the selection based on $griz$ bands provides excellent results onreal external datasets. The $griz$ selection is also purer at high redshift andmore complete at low redshift compared to colour selections found in theliterature. We find excellent agreement in terms of purity and completenessbetween the analysis of an independent, simulated Euclid galaxy catalogue andour calibration sample, except for galaxies at high redshifts, for which weobtain up to 50 percent points higher completeness. The combination of colourand photo-$z$ selections applied to simulated Euclid data yields up to 95%completeness, while the purity decreases down to 92% at high $z_{\rm l}$. Weshow that the calibrated colour selections provide robust results even whenobservations from a single band are missing from the ground-based data.Finally, we show that colour selections do not disrupt the shear calibrationfor stage III surveys.

Euclid Preparation. XXXVII. Galaxy colour selections with Euclid and ground photometry for cluster weak-lensing analyses