On matching cover of graphs

A k-matching cover of a graph $$G$$G is a union of $$k$$k matchings of $$G$$G which covers $$V(G)$$V(G). The matching cover number of $$G$$G, denoted by $$mc(G)$$mc(G), is the minimum number $$k$$k such that $$G$$G has a $$k$$k-matching cover. A matching cover of $$G$$G is optimal if it consists of $$mc(G)$$mc(G) matchings of $$G$$G. In this paper, we present an algorithm for finding an optimal matching cover of a graph on $$n$$n vertices in $$O(n^3)$$O(n3) time (if use a faster maximum matching algorithm, the time complexity can be reduced to $$O(nm)$$O(nm), where $$m=|E(G)|$$m=|E(G)|), and give an upper bound on matching cover number of graphs. In particular, for trees, a linear-time algorithm is given, and as a by-product, the matching cover number of trees is determined.