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Let $$G=(V,E)$$ be a finite or locally finite connected weighted graph, $$\Delta $$ be the usual graph Laplacian. Using heat kernel estimates, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on G We conclude that, for a graph satisfying curvature dimension condition $$\textit{CDE}'(n,0)$$ and $$V(x,r)\simeq r^m$$ , if $$02$$ , then there is a non-negative global solution u provided that the initial value is small enough. In particular, these results apply to the lattice $${\mathbb {Z}}^m$$ . [ABSTRACT FROM AUTHOR]