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signed graphs; $(k,d)$-graceful signed graphs
A $(p,q)$-sigraph $S$ is an ordered pair $(G,s)$ where $G = (V,E)$ is a $(p,q)$-graph and $s$ is a function which assigns to each edge of $G$ a positive or a negative sign. Let the sets $E^+$ and $E^-$ consist of $m$ positive and $n$ negative edges of $G$, respectively, where $m + n = q$. Given positive integers $k$ and $d$, $S$ is said to be $(k,d)$-graceful if the vertices of $G$ can be labeled with distinct integers from the set $\lbrace 0,1,\dots , k + (q-1)d\rbrace $ such that when each edge $uv$ of $G$ is assigned the product of its sign and the absolute difference of the integers assigned to $u$ and $v$ the edges in $E^+$ and $E^-$ are labeled $k, k + d, k + 2d,\dots , k + (m - 1)d$ and $-k, -(k + d), -(k + 2d),\dots ,-(k + (n - 1)d)$, respectively. In this paper, we report results of our preliminary investigation on the above new notion, which indeed generalises the well-known concept of $(k,d)$-graceful graphs due to B. D. Acharya and S. M. Hegde.
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