Generalize the vertex-cover lower bound beyond regular graphs

Generalize the lower bound $B_3(G) extgreatereq |K|$ from unswappable $k$-regular graphs to suitable non-$k$-regular graphs, either by removing or weakening the regularity hypothesis or by adapting the proof to specific nonregular graph families.

Background

Theorem 2.5 derives a bond-edge-type lower bound from a minimum vertex cover for unswappable graphs, while the paper’s upper-bound theorem is stated for k-regular graphs. The proof uses regularity in the counting argument that converts edge information into vertex-count information. Because this assumption appears only briefly, the authors explicitly identify extending the lower-bound result to nonregular graphs as an unresolved problem.