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.
In our earlier paper [9], generalizing the well known notion of graceful graphs, a $(p,m,n)$-signed graph $S$ of order $p$, with $m$ positive edges and $n$ negative edges, is called graceful if there exists an injective function $f$ that assigns to its $p$ vertices integers $0,1,\dots ,q = m+n$ such that when to each edge $uv$ of $S$ one assigns the absolute difference $|f(u) - f(v)|$ the set of integers received by the positive edges of $S$ is $\lbrace 1,2,\dots ,m\rbrace $ and the set of integers received by the negative edges of $S$ is $\lbrace 1,2,\dots ,n\rbrace $. Considering the conjecture therein that all signed cycles $Z_k$, of admissible length $ k \ge 3$ and signed structures, are graceful, we establish in this paper its truth for all possible signed cycles of lengths $ 0,2$ or $3\hspace{4.44443pt}(\@mod \; 4)$ in which the set of negative edges forms a connected subsigraph.