On magic and consecutive antimagic factorizations of graphs

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Let G and H be two graphs. An H-packing (resp. decomposition) of G is a family of subgraphs of G such that each edge of G belongs to at most (resp. exactly) one of the subgraphs and all subgraphs are isomorphic to H. An H-factorization of G is an H-decomposition of G if H is a factor of G. An H-packing (resp. decomposition) of G, say Ɓ, is called a (d, H)-packing (resp. decomposition) if there exists a bijection f from V (G) ∪ E(G) onto {1, 2,…,|V(G) ∪ E(G)|} such that {w(B)|B ∈Ɓ} = {a, a + d,…, a + (b − 1)d}, where w(B) is the sum of all vertex and edge labels on B (under f) and b = |Ɓ|. A (0, H)- and a (1, H)-packing (resp. decomposition) are said to an H-magic and an H-consecutive antimagic packing (resp. decomposition), respectively. The goal of this paper is to study an H-magic and an H-consecutive antimagic packing (resp. decomposition) of some regular graphs where H is a factor of the regular graph.