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, 14with 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.