Two-Tone Factors in Regular and Claw-Free Graphs: Existence, Algorithmic Construction, and Symmetry Breaking

Regular graphs are classical symmetric structures in graph theory, where each vertex has identical degree and the overall topology often exhibits strong automorphism properties. However, practical systems frequently require heterogeneous constraints, which can be modeled by introducing vertex colorings and non-uniform degree requirements, leading to controlled symmetry breaking. In this paper, we investigate two-tone factors in edge-connected regular graphs and claw-free cubic graphs under arbitrary red-blue vertex colorings. Using the framework of parity (g,f)-factors, we establish two main existence results. First, we prove that every λ-edge connected r-regular graph admits a two-tone ({k},{k,k+2})-factor for any coloring, provided that r/λ⩽k⩽r−r/λ, k⩽r−2, and k|G| is even. Second, we show that every 3-edge connected claw-free cubic graph admits a two-tone ({0,1},{2,3})-factor regardless of the coloring configuration. Beyond existence, we provide a constructive algorithm by reducing the parity factor problem to an exact f-factor problem and further to a perfect matching problem via vertex-splitting techniques. We rigorously justify the correctness of this reduction and show that the desired factor can be computed in polynomial time. From a structural perspective, our results reveal that edge-connectivity serves as a stabilizing mechanism that preserves parity feasibility under arbitrary color-induced perturbations, while claw-free constraints enforce local density that prevents parity imbalance. This provides a symmetry-based interpretation of two-tone factors as a balance between global regularity and local asymmetry. These findings contribute to both the theoretical development of factor theory and its algorithmic realization, with potential implications for deterministic network design and resource allocation in structured systems.