Hovey introduced A-cordial labelings in [4] as a simultaneous generalization of cordial and harmonious labelings. If A is an abelian group, then a labeling f: V(G) → A of the vertices of some graph G induces an edge-labeling on G; the edge uv receives the label f(u) + f(v). A graph G is A-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one. Research on A-cordiality...

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 $\{0,1,\cdots ,k+(q-1\left)d\right\}$ 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...

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 $0,1,\cdots ,q=m+n$ such that when to each edge $uv$ of $S$ one assigns the absolute difference $\left|f\right(u)-f(v\left)\right|$ the set of integers received by the positive edges of $S$ is $\{1,2,\cdots ,m\}$ and the set of integers received by the negative edges of $S$ is $\{1,2,\cdots ,n\}$. Considering the conjecture therein that all...