Characterization of graphs in eB_d minus A_d for d=2,3

Completely characterize all graphs G in the class eB_d \ A_d for d=2,3, where eB_d is the class of bipartite graphs defined in the paper and A_d consists of graphs in eB_d with exactly d components, each component being a one-point union of cycles.

Background

The paper studies bipartite graphs in the classes B_d and eB_d that admit local antimagic 2-colorings. The subclasses A_d comprise disconnected graphs with exactly d components, each component a one-point union of cycles. The paper completely characterizes A_d for d=2 and d=3, while earlier work and the present results provide the relevant classifications for these component structures.

For d=2 and d=3, Theorems 2.1 and 3.1 characterize the cases covered by the paper. The authors state that Problem 1.2—complete characterization of all graphs in eB_d—will be resolved for these values of d once the remaining graphs in eB_d\A_d are characterized. Such graphs have at most d components but are not among the exactly d-component one-point unions of cycles treated in the paper.