The number of vertices missed by a maximum matching in a graph G is the multiplicity of zero as a root of the matchings polynomial μ(G,x) of G, and hence many results in matching theory can be expressed in terms of this multiplicity. Thus, if mult(θ,G) denotes the multiplicity of θ as a zero of μ(G,x), then Gallai’s lemma is equivalent to the assertion that if mult(θ,G\u) < mult(θ,G) for each v...

A paired dominating set of a graph G without isolated vertices is a dominating set of G whose induced subgraph has a perfect matching. The paired domination number γpr(G) of G is the minimum cardinality amongst all paired dominating sets of G. The graph G is paired domination edge-critical (γprEC) if for every e ∈ E(G), γpr(G+ e) < γpr(G). We investigate the diameter of γprEC graphs. To this ef...

In this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let H be a set of connected graphs, each of which has three or more vertices. A graph G is said to beH-free if no graph inH is an induced subgraph of G. We completely characterize the setH such that every connected H-free graph of sufficiently large even order has a perfect matching in the foll...

For any graph H, let Forb∗(H) be the class of graphs with no induced subdivision of H. It was conjectured in [A.D. Scott, Induced trees in graphs of large chromatic number, Journal of Graph Theory, 24:297– 311, 1997] that, for every graph H, there is a function fH : N → R such that for every graph G ∈ Forb∗(H), χ(G) ≤ fH(ω(G)). We prove this conjecture for several graphs H, namely the paw (a tr...

Recently it was proposed that matching of global charges induced in vacuum by slowly varying, topologically non-trivial scalar fields provides consistency conditions analogous to the ’t Hooft anomaly matching conditions. We study matching of induced charges in supersymmetric SU(N) gauge theories with quantum modified moduli space. We find that the Wess–Zumino term should be present in the low e...

We provide a formula for the number of edges of a maximum induced matching in a graph. As applications, we give some structural properties of (k + 1 )K2-free graphs, construct all 2K2-free graphs, and count the number of labeled 2K2-free connected bipartite graphs.

A matching is a set of edges in graph with no common endpoint. M called acyclic if the induced subgraph on endpoints acyclic. Given G and an integer k, Acyclic Matching problem seeks for size k G. The known to be NP-complete. In this paper, we investigate complexity different aspects. First, prove that remains NP-complete class planar bipartite graphs maximum degree three arbitrarily large girt...

In a proper edge-coloring the edges of every color form matching. A matching is induced if end-vertices its induce strong an in which We consider intermediate types edge-colorings, where some colors are allowed to matchings, and remaining matchings. Our research motivated by conjecture proposed recent paper Gastineau Togni on S-packing edge-colorings (On cubic graphs, Discrete Appl. Math. 259 (...