On Super $(a,1)$-Edge-Antimagic Total Labelings of Grids and Crowns

Abstract

A graph $G(V,E)$ with order $p$ and size $q$ is called $(a,d)$-edge-antimagic total labeling graph if there exists a bijective function $f:V(G)\cup E(G)\to \{1,2,\dots ,p+q\}$ such that the edge-weights ${\lambda }_f(uv)=f(u)+f(v)+f(uv)$, $uv\in E(G)$, form an arithmetic sequence with first term $a$ and common difference $d$. Such a labeling is called super if the $p$ smallest possible labels appear at the vertices. In this paper, we study super $(a,1)$-edge-antimagic properties of $m(P_4\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{P}_n)$ for $m,n\ge 1$ and $m(C_n\odot \overline{K_l})$ for $n$ even and $m,l\ge 1$.