2-Power Unicyclic Cubic Graphs Revisited: A Correction and the Minimum Number of Power-of-Two Cycles

Pirzada, Shah and Baskoro (Electron. J. Graph Theory Appl. 10 (2022) 337–344) claimed to construct an infinite family of cubic graphs containing exactly one cycle whose length is a power of two, and, as a consequence, a proof of the Erdős–Gyárfás conjecture. This note shows that both claims fail: the 8-vertex building block of the construction does not exist (exhaustive check with witnesses), and no admissible block of order up to 20 is free of power-of-two cycles (16,474,007 graphs). The same computations yield structural bounds: a cubic graph without 2-power cycles has no edge cut of size at most two unless its order is at least 42, so every cubic counterexample on at most 40 vertices is 3-connected, and its cycle-separating 3-edge-cuts are confined to four explicit exceptional fragments. Complementing the correction, the minimum number f(n) of 2-power cycles over all connected cubic graphs of order n is computed exactly for all even n up to 22 (7,875,918 graphs): the minimum is 2, attained only at n = 14; in particular no 2-power unicyclic cubic graph of order at most 22 exists. The package contains the paper, all programs (C++ exact cycle counter, generation scripts), and complete witness data.