The Λ-Spectrum of Cubic Graphs: Perfect Matching Exchange and the Erdos-Gyárfás

We introduce the Λ-spectrum of a graph G with perfect matchings, defined as the set ofcycle lengths arising in the symmetric difference of pairs of perfect matchings. For bridgelesscubic graphs, this algebraic invariant provides a new lens through which to study cyclestructure and its connection to the Erdos-Gyárfás conjecture. We establish fundamentalproperties of Λ(G), prove that Λ(Petersen) = {8}, and demonstrate a striking dichotomybetween bipartite and non-bipartite cubic graphs: while all tested non-bipartite cubic graphssatisfy Λ(G)∩{4, 8, 16, . . .} ̸= ∅, the bipartite Heawood graph has Λ(Heawood) = {6, 10, 14}with no power of two.For the Flower Snark family Jn, we develop crossing parity sector markers explicit Z2-valued quadratic forms that partition the perfect matchings into two sectors of equal sizebased on crossing parity relative to a fixed vertex ordering. We observe computationallythat the coupling structure of these markers exhibits universal asymmetry: the edge a0b0serves as a hub with maximal degree in the coupling graph, while the last two petals havezero quadratic participation. This asymmetry appears to be intrinsic (gauge-invariant) andscales as O(n2) with the number of petals.Our computational results on over 60 cubic graphs provide strong evidence for a Λ-Erdos-Gyárfás conjecture for non-bipartite graphs, while identifying characteristic-2 arithmetic asa potential source of bipartite obstructions.