A sharp complement signless Laplacian spectral condition for odd [1, b]-factors

Abstract

We set G to be a connected graph of even order n ≥ b + 3, where b ≥ 3 is odd. We establish a sharp sufficient condition for the existence of an odd [1, b]-factor in terms of the signless Laplacian spectral radius q(G¯) of the complement graph. Specifically, if q(G¯)≤Φ(n,b)=n+2b−1+(n+2b−1)2−8b(b+1)2, then G has an odd [1, b]-factor, except for the unique extremal graph K1∨(Kn−b−2∪(b+1)K1). The equality case shows that the bound is best possible. We further formulate the theorem as a one-sided spectral screening procedure and illustrate its behavior through extremal perturbations, random graphs, and exact checks based on Amahashi’s criterion. A small sensor-network peer-checking example is included to show how the certificate can guarantee the feasibility of a spanning checking subnetwork under odd-degree and capacity constraints.