For a simple graph G, a graceful antimagic injection f in [0, M] is a map \(f: V(G)\rightarrow [0, M]\) such that the induced edge labels are different for all distinct edges and the vertex weights are different for all distinct vertices, where the induced edge label of uv is \(f^*(uv)=|f(u)-f(v)|\) and the vertex weight of v is \(\omega (v)=\sum _{uv\in E(G)}f^*(uv)\) . If \(\mathcal {Y}\) is a class of graphs of order n and size m not containing \(\bigcup K_2\) , \(GA(\mathcal {Y},n,m)\) denotes the smallest integer M such that every \(G\in \mathcal {Y}\) admits a graceful antimagic injection in [0, M]. For \(\mathcal {G}_\Delta \subset \mathcal {Y}\) with a maximum degree \(\Delta \) , we utilize Alon’s Combinatorial Nullstellensatz to find an upper bound for \(GA(\mathcal {G}_\Delta ,n,m)\) , that is, \(GA(\mathcal {G}_\Delta ,n,m)\) \(\le n+p(m-p)+\Delta (2n-4)\) , where \(p=\min \{\Delta ,\lfloor \frac{n}{2}\rfloor \}\) . We also prove that if \(\mathcal {T}_q \subset \mathcal {Y}\) is a class of trees whose base tree has q inner vertices, then \(GA(\mathcal {T}_q,n,n-1) \le 2n+q-3\) . We introduce a variant of the previous notion of injection, that is, the graceful \(\mathbb {D}\) -antimagic injection. Let G be a simple graph with diameter \({\textrm{diam}}(G)\) . Suppose that \(\mathbb {D}\) is a nonempty subset of the distance set \(\{0,1,\) \(\ldots , {\textrm{diam}}(G)\}\) and the \(\mathbb {D}\) -neighborhood of a vertex v, \(N_{\mathbb {D}}(v)\) , is the set of all vertices u with \(d(u,v)\in \mathbb {D}\) . A graceful \(\mathbb {D}\) -antimagic injection f in [0, N] is a map \(f: V(G)\rightarrow [0,N]\) such that all induced edge labels are distinct and all \(\mathbb {D}\) -vertex weights are also distinct, where the \(\mathbb {D}\) -vertex weight of v is the sum of the labels of the vertices in \(N_{\mathbb {D}}(v)\) . If \(\mathcal {X}\) is a class of graphs of order n and size m such that no two vertices have the same \(\mathbb {D}\) -neighborhood, the smallest possible N such that every graph in \(\mathcal {X}\) has a graceful \(\mathbb {D}\) -antimagic injection is denoted by \(G\mathbb {D}A(\mathcal {X},n,m)\) . Let the \(\mathbb {D}\) -degree of a vertex v be the cardinality of \(N_{\mathbb {D}}(v)\) . For \(\mathcal {G}_{\Delta ,\Delta _{\mathbb {D}}} \subset \mathcal {X}\) with a maximum degree \(\Delta \) and a maximum \(\mathbb {D}\) -degree \(\Delta _{\mathbb {D}}\) , we prove that \(G\mathbb {D}A(\mathcal {G}_{\Delta ,\Delta _{\mathbb {D}}},n,m) \le n+t(n-t)+p(m-p)\) , where \(t=\min \{\lfloor \frac{n}{2}\rfloor , \Delta _{\mathbb {D}}\}\) and \(p=\min \{\lfloor \frac{m}{2}\rfloor ,\Delta \}\) . Furthermore, if \(\mathcal {T}_k \subset \mathcal {X}\) is a class of trees with k leaves and \(\mathbb {D}=\{1\}\) , we prove that \(G\mathbb {D}A(\mathcal {T}_k,n,n-1) \le \max \{4n-6k-2, 3n-2k-3\}\) . Moreover, if \(\mathcal {T}_q \subset \mathcal {X}\) is a class of trees whose base tree has q inner vertices and \(\mathbb {D}=\{0,1\}\) , we prove that \(G\mathbb {D}A(\mathcal {T}_q,n,n-1) \le 2n+q-3\) .
