Matching Extendabilities of G = Cm ∨ Pn

A graph is considered to be induced-matching extendable (bipartite matching extendable) if every induced matching (bipartite matching) of G is included in a perfect matching of G. The induced-matching extendability and bipartite-matching extendability of graphs have been of interest. By letting G = C m ∨ P n ( m ≥ 3 and n ≥ 1 ) be the graph join of C m (the cycle with m vertices) and P n (the path with n vertices) contains a perfect matching, we find necessary and sufficient conditions for G to be induced-matching extendable and bipartite-matching extendable.

Extending a perfect matching to a Hamiltonian cycle

Graph Theory International audience Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.

K − Bipartite Matching Extendability of Special Graphs

Let be a simple connected graph containing a perfect matching. For a positive integer , , is said to be bipartite matching extendable if every bipartite matching of with is included in a perfect matching of . In this paper, we show that bipartite matching extendability of some special graphs.

BIPARTITE MATCHING EXTENDABILITY AND TOUGHNESS

Let G be a simple graph containing a perfect matching. G is said to be bipartite matching extendable (BM-extendable) if every matching M which is a perfect matching of an induced bipartite subgraph extends to a perfect matching of G. In this paper, we study some relations between toughness and BM-extendability of a graph, including some sufficient or necessary conditions about toughness for a graph to be BM-extendable, and a sufficient condition for a BM-extendable graph to be 1-tough.