General existence and classification of copies with prescribed local antimagic chromatic number

Establish that for every connected graph G of order n≥3 without pendant vertices, there exist graphs mG such that χ_la(mG)=mn−k for every m≥2 and every 0≤k≤mn−2, and characterize all connected graphs G with this property.

Background

The paper relates its results to an earlier theorem proving existence of connected graphs with pendant vertices whose local antimagic chromatic number attains prescribed values. It then considers disjoint unions of copies of connected graphs without pendant vertices.

The conjecture proposes a broad extension: for every admissible number of copies m and every target deficit k, some graph formed by m disjoint copies of a connected graph G should have local antimagic chromatic number mn−k. It also asks for a complete characterization of all such connected graphs. The authors note that their Corollaries 2.3 and 3.3 verify the conjecture only for m=2,3 and the extremal case k=mn−2.