Maximally matched and super matched regular graphs

Perfect matchings in highly cyclically connected regular graphs

“…For k = 1, we are able to reformulate Theorem 2 as follows. This result is closely related to Theorem 3.5 in [15]. Theorem 3.…”

“…Theorem 1 has been examined in various modifications and generalisations. Instead of assuming that all edge-cuts have size at least d − 1 one needs to assume it only for odd edgecuts [8,15]. As if we remove an odd edge-cut from the graph the resulting graph does not admit a perfect matching, Cruse's result holds in the converse direction.…”

Fractional matching preclusion number of graphs and the perfect matching polytope

“…Noting that for each v ∈ V (G), there is a row in M G corresponding to incidence vector of ∂ G (v), we denote this row vector by ∂ G (v). In [21], we introduced a 0-1 linear programming for matching preclusion number of G. Let G be a graph with an even number of vertices. We denote M(G) be the set consisting of its all perfect matchings and y be a vector in R E(G) .…”

“…In [21], by applying this 0-1 linear programming on r-regular graph G we showed that mp(G) = r if and only if each non-trivial odd cut of G has at least r edges.…”

“…Since matching preclusion problem was proposed, it has been studied for many graphs, such as hypercube [3], k-ary n-cube [27], tori network [10], balanced hypercube [23], folded Petersen cube [5], cube-connected cycle [20] and pancake and burnt pancake graph [8,17]. Furthermore, there are also some papers studying matching preclusion for some classes of graph, such as bipartite graph [6,7], regular graph [9,21], vertex-transitive graph [19] and Cartesian product of graphs [10,21]. In complexity issue, M. Lacroix et al [18] showed that matching preclusion problem is NP-complete even for bipartite graphs.…”