A well known theorem of Delmotte is that Gaussian bounds, parabolic Harnack inequality, and the combination of volume doubling and Poincaré inequality are equivalent for graphs. In this talk, we consider graphs for which these conditions hold, but only for sufficiently large balls, and show a similar equivalence. We also show more precise sufficient conditions on the range of balls for which good behaviour is required in order to obtain heat kernel bounds in a fixed ball.