We introduce and study a new four-peg variant of the Tower of Hanoi problem under parity constraints. Two pegs are neutral and allow arbitrary disc placements, while the remaining two pegs are restricted to discs of a prescribed parity: one for even-labelled discs and the other for odd-labelled discs. Within this constrained setting, we investigate four canonical optimization objectives according to distinct target configurations and derive for each the exact number of moves required to optimally transfer the discs. We establish a coupled system of recursive relations governing the four optimal move functions and unfold them into higher-order recurrences and explicit closed forms. These formulas exhibit periodic growth patterns and reveal that all solutions grow exponentially, but at a significantly slower rate than the classical three-peg case. In particular, each optimal move sequence has an a half-exponential-like asymptotic order induced by the parity restriction. In addition, we define the associated parity-constrained Hanoi graph, whose vertices represent all feasible states and whose edges represent legal moves. We determine its order, degrees, connectivity, planarity, diameter, Hamiltonicity, clique number, and chromatic properties, and show that it lies strictly between the classical three- and four-peg Hanoi graphs via the inclusion relation.

翻译：本文引入并研究了一种在奇偶约束下的新型四柱汉诺塔问题变体。其中两根柱子为中性柱，允许任意圆盘放置；而剩余两根柱子则受限于特定奇偶性的圆盘：一根仅允许偶数编号圆盘，另一根仅允许奇数编号圆盘。在此约束条件下，我们针对四种典型的目标配置优化目标展开研究，推导出每种情况下最优转移圆盘所需的确切移动步数。我们建立了一个耦合的递归关系系统来描述四种最优移动函数，并将其展开为高阶递推关系和显式闭式解。这些公式呈现出周期性增长模式，并揭示所有解均呈指数增长，但增长速度显著慢于经典三柱情形。特别地，每个最优移动序列均具有由奇偶约束诱导的半指数级渐近阶。此外，我们定义了关联的奇偶约束汉诺图，其顶点表示所有可行状态，边表示合法移动。我们确定了该图的阶数、度、连通性、可平面性、直径、哈密顿性、团数及染色性质，并通过包含关系证明其严格介于经典三柱与四柱汉诺图之间。