The signed edge domination number of a graph is an edge variant of the signed domination number. The closed neighbourhood $N_G\left[e\right]$ of an edge $e$ in a graph $G$ is the set consisting of $e$ and of all edges having a common end vertex with $e$. Let $f$ be a mapping of the edge set $E\left(G\right)$ of $G$ into the set $\{-1,1\}$. If ${\sum }_{x\in N\left[e\right]}f\left(x\right)\ge 1$ for each $e\in E\left(G\right)$, then $f$ is called a signed edge dominating function on $G$. The minimum of the values ${\sum }_{x\in E\left(G\right)}f\left(x\right)$, taken over all signed edge dominating function $f$ on $G$, is called the signed edge domination number...

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...

A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = E₁,...,Eₘ, together with integers $s_i$ and $t_i$ satisfying $1\le s_i\le t_i\le \left|E_i\right|$ for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge $E_i$ satisfies $s_i\le \left|\phi \left(E_i\right)\right|\le t_i$. The hypergraph ℋ is colorable if it admits at least one proper coloring. We consider hypergraphs ℋ over a “host graph”, that means a graph G on the same vertex set X as ℋ, such that each $E_i$ induces a connected subgraph in G....

A graph $G$ is a $\{d,d+k\}$-graph, if one vertex has degree $d+k$ and the remaining vertices of $G$ have degree $d$. In the special case of $k=0$, the graph $G$ is $d$-regular. Let $k,p\ge 0$ and $d,n\ge 1$ be integers such that $n$ and $p$ are of the same parity. If $G$ is a connected $\{d,d+k\}$-graph of order $n$ without a matching $M$ of size $2\left|M\right|=n-p$, then we show in this paper the following: If $d=2$, then $k\ge 2(p+2)$ and (i) $n\ge k+p+6$. If $d\ge 3$ is odd and $t$ an integer with $1\le t\le p+2$, then (ii) $n\ge d+k+1$ for $k\ge d(p+2)$, (iii) $n\ge d(p+3)+2t+1$ for $d(p+2-t)+t\le k\le d(p+3-t)+t-3$, (iv) $n\ge d(p+3)+2p+7$ for $k\le p$. If $d\ge 4$ is even, then (v) $n\ge d+k+2-\eta$ for $k\ge d(p+3)+p+4+\eta$, (vi) $n\ge d+k+p+2-2t=d(p+4)+p+6$ for $k=d(p+3)+4+2t$ and $p\ge 1$,...