\((a,d)\)-Antimagic Parachutes \(II\)

In \([2, 3]\), the authors dealt with the problem of determining the set \(\Gamma(G)\) of all \((a, d)\)-antimagic graphs, \(a, d \in \mathbb{N}\), where the concept of an \((a, d)\)-antimagic graph is a variation of the concept of an antimagic graph given in [4]. A connected graph \(G = (V, E) \in \Gamma=\) set of all finite undirected graphs without loops and multiple edges on \(n = |V| \geq 3\) vertices and \(m = |E| \geq 2\) edges is said to be \((a, d)\)-antimagic iff its edges can be assigned mutually distinct nonnegative integers from \(\{1, 2, \ldots, m\}\) so that the values of the vertices obtained as the sums of the numbers assigned to the edges incident to them can be arranged in the arithmetic progression \(a, a + d, \ldots, a + (n – 1)d\). In [2], the authors obtained some interesting general results on \((a, d)\)-antimagic graphs from \(\Gamma(G)\) by applying the theory of linear Diophantine equations and other number theoretical topics. Applying these general results to wheels \(W_{g , b} = 1 \ast C_{g + b}, g \geq 3, b \geq 1, C_{g + b} =\) cycle of order \(g + b\), and parachutes \(P_{g, b}\) as the spanning subgraph of \(W_{g+b}\) arising from \(W_{g+b}\) by removing \(b\) successive spokes of \(W_{g+b}\), we succeeded in proving that every wheel \(W_{g+b}\) cannot be \((a, d)\)-antimagic and, for every \(g \geq 3\) or \(g \geq 4\), there are the five integers \(b_1 = 2g^2 – 3g – 1, b_2 = g^2 – 2g – 1, b_3 = g – 1, b_4 = g – 3\) and \(b_5 = \frac{1}{2}(g^2 – 3g – 2)\) with the property that the corresponding parachute \(P_{g, b_i}, i = 1, 2, \ldots, 5\), can be \((a, d)\)-antimagic. If \(\Gamma_i(P)\) denotes the set \(\Gamma_i(P)= \{P_{g, b_i} \in \Gamma(P) \mid g \geq 3\}, i = 1, 2, \ldots, 5\), the main result in [2] says that \(\Gamma_3(P) = \{P_{3, 2}, P_{4, 3},\ldots, P_{8,7}, P_{10,9}, P_{11, 10}\}\) and \(\Gamma_4(P) = \{P_{4, 1}, P_{5, 2}, \ldots, P_{10, 7}\}\). Concerning \(\Gamma_1(P), \Gamma_2(P)\) and \(\Gamma_5(P)\) the authors conjecture that they are infinite. Here, we continue [2] and prove the conjecture given in [2] for \(\Gamma_1(P)\) and \(\Gamma_2(P)\). Instead of \(\Gamma(P)\) we prove the infiniteness of \(\Gamma'(P) = \{P_g,\frac{1}{3} (2g^2 – 5g – 3) \in \Gamma(P) \mid g \equiv 0(3) \text{ or } g \equiv 1(3)\}\). Furthermore, we succeed in showing the existence of integers \(b_{min} \in \{\frac{g^2 – 3g – 2}{2}, \frac{g^2 – 4g – 3}{3},\frac{g^2 – 5g – 4}{4}, \frac{2g^2 – 7g – 5}{5}, \}\) with respect to \(g \geq 26\) with the property that the parachute \(P_{g, b}\) is not \((a, d)\)-antimagic for each positive integer \(b \leq b_{min}\). The immediate consequence of this fact is that for every \(g \geq 26\) there are at most \(8\) different integers \(b \geq b_{min}\) such that the corresponding parachute \(P_{g, b}\) could be \((a, d)\)-antimagic.