On Vertex Magic 3-Regular Graphs with a Perfect Matching

Let $G=(V,E)$ be a finite simple graph with $p=\left|V\right|$ vertices and $q=\left|E\right|$ edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection f from $V\cup E$ to the consecutive integers$1,2,\dots ,p+q,$ with the property that, for every vertex $u\in V$, one has $f\left(u\right)+{\sum }_{uv\in E}f\left(uv\right)=k$ for some magic constant k. The vertex magic total labeling is called E-super if furthermore $f\left(E\right)=\{1,2,\dots ,q\}$. A graph is called (E-super) vertex magic if it admits an (E-super) vertex magic total labeling. In this paper, we verify the existence of E-super vertex magic total labeling for a class of 3-regular graphs with a perfect matching, and we confirm the existence of such a labeling for general regular graphs of odd degree containing particular classes of 3-factors, which provides us with known and new examples. Note that Harary graphs are among the popular models used in communication networks. In 2012, G. Marimuthu and M. Balakrishnan raised a conjecture that if $n>4$, $n\equiv 0\left(mod4\right)$ and m is odd, then the Harary graph $H_{m,n}$ admits an E-super vertex magic labeling. Among others, we are able to verify this conjecture except for one case while $m=3$ and $n\equiv 4\left(mod8\right)$.