\(H\)-coverings of path-amalgamated ladders and fans (Q2214853)

Summary: Let \(\mathbb{G}\) be a connected, simple graph with finite vertices \(v\) and edges \(e\). A family \(\left\{ \mathbb{G}_1, \mathbb{G}_2, \ldots, \mathbb{G}_p\right\}\subset\mathbb{G}\) of subgraphs such that for all \(e\in E\), \(e\in \mathbb{G}_l\), for some \(l\), \(l=1,2,\ldots,p\) is an edge-covering of \(\mathbb{G} \). If \(\mathbb{G}_l\cong\mathbb{H}\), \(\forall l \), then \(\mathbb{G}\) has an \(\mathbb{H} \)-covering. Graph \(\mathbb{G}\) with \(\mathbb{H} \)-covering is an \(\left( a_d, d\right)\)-\(\mathbb{H} \)-antimagic if \(\psi:V\left( \mathbb{G}\right)\cup E\left( \mathbb{G}\right)\longrightarrow\left\{ 1,2, \ldots, v + e\right\}\) a bijection exists and the sum over all vertex-weights and edge-weights of \(\mathbb{H}\) forms a set \(\left\{ a_d, a_d + d, \ldots, a_d + \left( p - 1\right) d\right\}\). The labeling \(\psi\) is super for \(\psi\left( V \left( \mathbb{G}\right)\right)=\left\{ 1,2,3, \ldots, v\right\}\) and graph \(\mathbb{G}\) is \(\mathbb{H} \)-supermagic for \(d=0\). This manuscript proves results about super \(\mathbb{H} \)-antimagic labeling of path amalgamation of ladders and fans for several differences.