### On total covering and matching of graphs

A Meir
1978 Journal of combinatorial theory. Series B (Print)
1. Let G be a graph with no loops or multiple edges having node set N(G) and edge set E(G). The elements of N(G) u E(G) are called elements of G. A node P is said to cover itself, all edges incident to P, and all nodes joined to P. An edge (P, Q) covers itself, the nodes P and Q, and all edges incident to P or Q. Two elements of G are called independent if neither one covers the other. A set % of elements of G is called a total cover if the elements of %Y cover all elements of G and V is
more » ... . A set 9 of elements of G is called total matching if the elements of 4 are pariwise independent and 4 is maximal. For a fixed graph G, let up(G) = inf / V 1, CQ'(G) = sup j V ), where the inf and sup are taken over all total covers 5%'. Similarly, let P2'(G) = inf j 4 i, ,4(G) = SUP i 3 I, where the inf and sup are taken over all total matchings X. These concepts and quantities were introduced in [I]. The rest&s of [l] are variations of earlier results of Gallai [a]. 2. In [I] it was shown that if G is a connected graph of order n > 2, then and n < 4G) + P,(G) < @n -1).