A Note on Signed k-Submatching in Graphs

Let $G$ be a graph of order $n$. For every $v\in V(G)$, let $E_G(v)$ denote the set of all edges incident with $v$. A signed $k$-submatching of $G$ is a function $f:E(G)⟶\{-1,1\}$, satisfying $f(E_G(v))\le 1$ for at least $k$ vertices, where $f(S)={\sum }_{e\in S}f(e)$, for each $ S\subseteq E(G)$\mathrm{.}Themaximumofthevalueof$f(E(G))$,takenoverallsigned$k$-submatching$f$of$G$,iscalledthesigned$k$-submatching number and is denoted by ${\beta }_S^k(G)$. In this paper, we prove that for every graph $G$ of order $n$ and for any positive integer $k\le n$, $\beta ^k_S (G) \geq n-k - \omega(G)$,where$w(G)$isthenumberofcomponentsof$G$. This settles a conjecture proposed by Wang. Also, we present a formula for the computation of ${\beta }_S^n(G)$.