Toughness, binding number and restricted matching extension in a graph

A connected graph G with at least 2m + 2n + 2 vertices is said to satisfy the property E(m, n) if G contains a perfect matching and for any two sets of independent edges M and N with |M| = m and |N| = n with M boolean AND N = empty set, there is a perfect matching F in G such that M subset of F and N boolean AND F = empty set. In particular, if G is E(m, 0), we say that G is m-extendable. One of the authors has proved that every m-tough graph of even order at least 2m + 2 is m-extendable (Plummer, 1988). Chen (1995) and Robertshaw and Woodall (2002) gave sufficient conditions on binding number for m-extendability. In this paper, we extend these results and give lower bounds on toughness and binding number which guarantee E(m, n). (C) 2016 Elsevier B.V. All rights reserved.