Given a function f : N → R, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k ≤ (n− f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f -connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-facto...

We generalize the well-known theorem of Corrádi and Hajnal which says that if a given graph G has at least 3k vertices and the minimum degree of G is at least 2k, then G contains k vertex-disjoint cycles. Our main result is the following; for any integer k, there is an absolute constant ck satisfying the following; let G be a graph with at least ck vertices such that the minimum degree of G is ...

Introduction: Nowadays, removable orthodontic appliances are widely used in the treatment of straightening teeth and guided the growth of the jaws. These devices have acrylic parts in which methyl methacrylate monomers are used and have been the subject of discussion in recent years as a cause of epithelial lesions in the oral cavity, and they are considered as one of the factors in mucosal al...

This paper is intended to investigate the effects of infected external computers and removable storage media on the spread of computer viruses. To this end, a new dynamical model, which has four compartments, is proposed. The analysis of the model shows that the unique equilibrium is globally asymptotically stable. This result is well suited for numerical implementation. An illustration of the ...

The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature. In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle and vertex k-colorability. In particular, we discover the ...

For a graph G = (V,E) and a vertex v ∈ V , let T (v) be a local trace at v, i.e. T (v) is an Eulerian subgraph of G such that every walk W (v), with start vertex v can be extended to an Eulerian tour in T (v). We prove that every maximum edge-disjoint cycle packing Z∗ of G induces a maximum trace T (v) at v for every v ∈ V . Moreover, if G is Eulerian then sufficient conditions are given that g...

Strictly Chordality-k graphs (SCk) are graphs which are either cycle-free or every induced cycle is of length exactly k, k ≥ 3. Strictly chordality-3 and strictly chordality-4 graphs are well known chordal and chordal bipartite graphs, respectively. For k ≥ 5, the study has been recently initiated in [1] and various structural and algorithmic results are reported. In this paper, we show that ma...

Let G be a graph with a nonnegative integral function w defined on V (G). A collection F of subsets of V (G) (repetition is allowed) is called a feedback vertex set packing in G if the removal of any member of F from G leaves a forest, and every vertex v ∈ V (G) is contained in at most w(v) members of F . The weight of a cycle C in G is the sum of w(v), over all vertices v of C. The purpose of ...

In an era of implants and fixed prosthesis, removable partial dentures are still a treatment of choice for a wide range of patients and clinicians. As every other treatment option, removable prosthesis also has some disadvantages. But with introduction of flexible partial dentures, many of the disadvantages of conventional acrylic and cast partial dentures are taken care of. This article discus...

Henning and Yeo [SIAM J. Discrete Math. 26 (2012) 687–694] conjectured that a 3-regular digraph D contains two vertex disjoint directed cycles of different length if either D is of sufficiently large order or D is bipartite. In this paper, we disprove the first conjecture. Further, we give support for the second conjecture by proving that every bipartite 3-regular digraph, which either possesse...