Li-Hui Wang

We study two-tone factors in edge-connected regular graphs and claw-free cubic graphs under red-blue vertex colorings. A two-tone factor is a spanning subgraph in which vertices of different colors are assigned different allowed degree sets. For edge-connected regular graphs, we prove an existence theorem for two-tone $\left(\right\{k\},\{k,k+2\left\}\right)$-factors under arbitrary red-blue colorings and describe a constructive factor-theoretic route to matching-type formulations. For the claw-free cubic setting, we show that the natural arbitrary-coloring extension for $\left(\right\{0,1\},\{2,3\left\}\right)$-factors is false in general. Nevertheless, we prove a restricted positive result when the graph has a triangle decomposition and the coloring is constant on each triangle, and we give a mask-consistency characterization for triangle-inflated cubic graphs. The computational section is therefore framed as an implementation and proof-audit study: it certifies recovered factors in the regular case, cross-checks claw-free cubic stress tests by independent MILP and mask-CSP formulations, and explains why only provable restrictions are retained as theorems. From the perspective of symmetry, the contrast identifies a boundary between symmetry-stable and symmetry-fragile colored degree constraints.