On H-antimagic coverings for m-shadow and closed m-shadow of connected graphs

Abstract

An ( a , d ) -H-antimagic total labeling of a simple graph G admitting an H-covering is a bijection φ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , | V ( G ) | + | E ( G ) | } such that for all subgraphs H ' of G isomorphic to H, the set of H ' -weights given by w t φ ( H ' ) = ∑ v ∈ V ( H ' ) φ ( v ) + ∑ e ∈ E ( H ' ) φ ( e ) forms an arithmetic sequence a , a + d , … , a + ( t - 1 ) d where a > 0 , d ⩾ 0 are two fixed integers and t is the number of all subgraphs of G isomorphic to H. Moreover, such a labeling φ is called super if the smallest possible labels appear on the vertices. A (super) ( a , d ) -H-antimagic graph is a graph that admits a (super) ( a , d ) -H-antimagic total labeling. In this paper the existence of super ( a , d ) -H-antimagic total labelings for the m-shadow and the closed m-shadow of a connected G for several values of d is proved.

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