On super vertex-graceful unicyclic graphs

Abstract

A graph G with p vertices and q edges, vertex set V(G) and edge set E(G), is said to be super vertex-graceful (in short SVG), if there exists a function pair (f, f ⁺) where f is a bijection from V(G) onto P, f ⁺ is a bijection from E(G) onto Q, f ⁺((u, v)) = f(u) + f(v) for any (u, v) ∈ E(G),

$$ Q = \left\{ \begin{gathered} \{ \pm 1, \ldots , \pm \tfrac{1} {2}q\} , if q is even, \hfill \\ \{ 0, \pm 1, \ldots , \pm \tfrac{1} {2}(q - 1)\} , if q is odd, \hfill \\ \end{gathered} \right. $$

and

$$ P = \left\{ \begin{gathered} \{ \pm 1, \ldots , \pm \tfrac{1} {2}p\} , if p is even, \hfill \\ \{ 0, \pm 1, \ldots , \pm \tfrac{1} {2}(p - 1)\} , if p is odd. \hfill \\ \end{gathered} \right. $$

We determine here families of unicyclic graphs that are super vertex-graceful.