Geodesic
On super vertex-graceful unicyclic graphs
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 1-22 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $(u, v) \in E(G)$, $$ Q = \begin{cases} \{\pm 1,\dots , \pm \frac 12q\},\text {if $q$ is even,}\\ \{0, \pm 1, \dots , \pm \frac 12(q-1)\},\text {if $q$ is odd,} \end{cases} $$ and $$ P = \begin{cases} \{\pm 1,\dots , \pm \frac 12p\},\text {if $p$ is even,}\\ \{0, \pm 1, \dots , \pm \frac 12(p-1)\},\text {if $p$ is odd.} \end{cases} $$ \endgraf We determine here families of unicyclic graphs that are super vertex-graceful.
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 $(u, v) \in E(G)$, $$ Q = \begin{cases} \{\pm 1,\dots , \pm \frac 12q\},\text {if $q$ is even,}\\ \{0, \pm 1, \dots , \pm \frac 12(q-1)\},\text {if $q$ is odd,} \end{cases} $$ and $$ P = \begin{cases} \{\pm 1,\dots , \pm \frac 12p\},\text {if $p$ is even,}\\ \{0, \pm 1, \dots , \pm \frac 12(p-1)\},\text {if $p$ is odd.} \end{cases} $$ \endgraf We determine here families of unicyclic graphs that are super vertex-graceful.
Classification : 05C78
Keywords: graceful; edge-graceful; super edge-graceful; super vertex-graceful; amalgamation; trees; unicyclic graphs 
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Lee, Sin-Min; Leung, Elo; Ng, Ho Kuen. On super vertex-graceful unicyclic graphs. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 1-22. http://geodesic.mathdoc.fr/item/CMJ_2009_59_1_a0/

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