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We study the one-loop effective action for $6D,$ ${\cal N}=(1,0)$supersymmetric Yang--Mills (SYM) theory with hypermultiplets and $6D,$ ${\calN}=(1,1)$ SYM theory as a subclass of the former, using the off-shellformulation of these theories in $6D,$ ${\cal N}=(1,0)$ harmonic superspace. Wedevelop the corresponding supergraph technique and apply it to compute theone-loop divergences in the background field method ensuring the manifest gaugeinvariance. We calculate the two-point Green functions of the gauge superfieldand the hypermultiplet, as well as the three-point gauge-hypermultipet Greenfunction. Using these Green functions and exploiting gauge invariance of thetheory, we find the full set of the off-shell one-loop divergent contributions,including the logarithmic and power ones. Our results precisely match withthose obtained earlier in [1,2] within the proper time superfield method.

Supergraph analysis of the one-loop divergences in $6D$, ${\cal N} = (1,0)$ and ${\cal N} = (1,1)$ gauge theories