This paper consists of five interesting lemmas and a theorem on certain antimagic regular graphs. The well-written paper makes for absolutely delightful reading. It contains many newly introduced, well-explained, well-illustrated notations and notions.

An antimagic labeling of a graph G=(V,E) is a bijective function f:E→ {1,2, . . . ,|E|} such that the sum of labels of edges incident on distinct vertices are distinct. An antimagic graph is a graph with an antimagic labeling.

A Y-link of a bipartite graph G=(X ∪ Y, E) is a 2-path with both end points in Y and Y-link of G is a family of vertex disjoint Y-links. Let v^* be a vertex of G and let L_i be the set of vertices in G, which are at a distance i from v^*. If σ denotes a mapping from V(G) - v^* to the edge set of G, with respect to which each vertex u ∈ L_i is assigned an edge σ(u) of G [L_i-1, L_i] incident to u, then the authors infer that σ(u) cannot be a linking edge for any u ∈ L_i. If G _σ[L_i-1, L_i] is the subgraph of G [L_i-1, L_i], obtained by deleting all the edges {σ(u):u ∈ L_i} and G’_σ[L_i-1, L_i] is the subgraph of G_σ [L_i-1, L_i] obtained by deleting all the linking edges, a bad component of G’_σ [L_i-1, L_i] is a 2-regular component H of it such that all vertices of H ∩ L_i are covered by linking edges. An L_i-link uvu’ in F_i is free, if one of u and u’ does not belong to any bad component of G_σ[L_i-1, L_i]. Invoking these notions, it is proven that G_σ[L_i-1, L_i] has a bad component, and implies F_i contains at least k free L_i-links. If n_i, a_i, b_i and c_i respectively denote the cardinalities of L_i, E(G[L_i]), E(G’_σ[L_i-1, L_i]), and E(G_σ[L_i-1, L_i]), then the authors proved that f(u_jv_j) ≤ d_i+a_i+b_i+c_i - k, for any link edge u_jv_j and u_j is a vertex in a bad component and the sum of labels of edges in E(u)-σ(u) is less than or equal to (2k+1)(d_i+a_i)+(k+1)b_i+c_i+k. Hence, the key result of the paper, which states that all (2k + 2)-regular graphs are antimagic for any integer k ≥ 1, follows immediately.