Super Vertex-magic Total Labelings of $W_{3,n}$

Abstract

A graph $G$ is called super vertex-magic total labelings if there exists a bijection $f$ from $V(G)\cup E(G)$ to $\{1,2,\dots ,|V(G)|+|E(G)|\}$ such that $f(v)+\sum _{u\sim v}f(vu)=C$, where the sum is over all vertices $u$ adjacent to $v$ and $f(V(G))=\{1,2,\dots ,|V(G)|\}$, $f(E(G))=\{|V(G)|+1,|V(G)|+2,\dots ,|V(G)|+|E(G)|\}$. $TheKnödelgraphs$ $W_{\mathrm{\Delta },n}$ have even $n\ge 2$ vertices and degree $\mathrm{\Delta }$, $1\le \mathrm{\Delta }\le \lfloor {\log }_{2}n\rfloor$. The vertices of $W_{\mathrm{\Delta },n}$ are the pairs $(i,j)$ with $i=1,2$ and $0\le i\le n/2-1$. For every $j$, $0\le j\le n/2-1$, there is an edge between vertex $(1,j)$ and every vertex $(2,(j+2^k-1)\mathrm{mod}(n/2))$, for $k=0,\dots ,\mathrm{\Delta }-1$. In this paper, we show that $W_{3,n}$ is super vertex-magic for $n\equiv 0\mathrm{mod}4$.