A note on signed k-submatching in graphs

Let G be a graph of order n. For every v ∈ V (G), let EG(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(EG(v)) ≤ 1 for at least k vertices, where f(S) = ∑ e∈S f(e), for each S ⊆ E(G). The maximum of the value of f(E(G)), taken over all signed k-submatching f of G, is called the signed k-submatching number and is denoted by β S(G). In this paper, we prove that for every graph G of order n and for any positive integer k ≤ n, β S(G) ≥ n− k−ω(G), where w(G) is the number of components of G. This settles a conjecture proposed by Wang. Also, we present a formula for the computation of β S(G). 2010 AMS Subject Classification: 05C70, 05C78.