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In this note we give a characterization of the complete bipartite graphs which have an even (odd) [a,b]-factor. For general graphs we prove that an a-edge connected graph G with n vertices and with δ(G) ≥ max{a+1,an/(a+b) + a - 2} has an even [a,b]-factor, where a and b are even and 2 ≤ a ≤ b. With regard to the edge-connectivity this result is slightly better than one of the similar results obtained by Kouider and Vestergaard in 2004 and unlike their results, this result has no restriction on the...

Let G be a graph. A function f : V (G) → {−1, 1} is a signed k- independence function if the sum of its function values over any closed neighborhood is at most k − 1, where k ≥ 2. The signed k-independence number of G is the maximum weight of a signed k-independence function of G. Similarly, the signed total k-independence number of G is the maximum weight of a signed total k-independence function of G. In this paper, we present new bounds on these two parameters which improve some existing bounds....

The open neighborhood $N_G\left(e\right)$ of an edge $e$ in a graph $G$ is the set consisting of all edges having a common end-vertex with $e$. Let $f$ be a function on $E\left(G\right)$, the edge set of $G$, into the set $\{-1,1\}$. If ${\sum }_{x\in N_G\left(e\right)}f\left(x\right)\ge 1$ for each $e\in E\left(G\right)$, then $f$ is called a signed edge total dominating function of $G$. The minimum of the values ${\sum }_{e\in E\left(G\right)}f\left(e\right)$, taken over all signed edge total dominating function $f$ of $G$, is called the signed edge total domination number of $G$ and is denoted by ${\gamma }_{st}^{\text{'}}\left(G\right)$. Obviously, ${\gamma }_{st}^{\text{'}}\left(G\right)$ is defined only for graphs $G$ which have no connected components...