Regular disconnected graphs with local antimagic chromatic number at least the chromatic number

Show the existence of infinitely many regular disconnected graphs without pendant vertices such that, for each integer k≥2, the chromatic number is k and the local antimagic chromatic number is at least k.

Background

The paper studies local antimagic edge labelings of bipartite graphs whose vertices have degree 2 except for exactly three vertices of degree at least 3, and derives necessary and sufficient structural conditions for such graphs to have local antimagic chromatic number 2. In its conclusion, the paper relates these results to known examples of disconnected graphs without pendant vertices whose local antimagic chromatic number exceeds their ordinary chromatic number.

The unresolved problem asks for an infinite family, for every prescribed integer k≥2, of regular disconnected graphs without pendant vertices having ordinary chromatic number exactly k while requiring at least k colors in every local antimagic vertex coloring. Establishing such families would extend the cited constructions involving disjoint unions of cycles and would clarify how regularity, disconnectedness, and local antimagic chromatic number interact.