Connectivity Properties of Generalized K4-Hypercubes

We consider the class of generalized hypercubes constructed recursively from the graph [Formula: see text] by repeatedly taking two copies of such a graph with a perfect matching added in between. We show that all graphs obtained this way have very good connectivity properties. They are all maximally connected, and even when linearly many vertices are deleted, the remaining graph will have a large connected component with only a few vertices in other components. We also show examples that we can delete more vertices in certain graphs in this class to get the second largest component to have certain sizes, including the case when we get two components of equal size. We conjecture that these examples are best possible.

Selfish Caching and Facsimile Allocation over a Mobile Ad Hoc Network

Mobile adhoc networks (MANETs) have drawn attention to multitudinous consideration because of the univerality of mobile devices as well as the developments in wireless era. MANET is a peer-to-peer multi hop cellular wireless era community which does not have both difficult and speedy infrastructure and a relevant server. Every vertex of a MANET performs like a router and communicates with every unique. There exist numerous information duplication strategies which were presented to reduce the execution squalor. All are concluded that everyone cell vertices cooperate completely from the perspective of sharing their memory vicinity. But, via a few methods few vertices might additionally behave selfishly and determine simplest to cooperate in part or never with different vertices. The selfish vertices ought to then lessen the overall information approachability within the network. From this work, we try to take a look at the influence of selfish vertices in a mobile ad hoc community in terms of reproduction issuance i.e Selfish nodes are dealt with in replica allocation.

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.

Triangulations with few vertices of manifolds with non-free fundamental group

We study lower bounds for the number of vertices in a PL-triangulation of a given manifold M. While most of the previous estimates are based on the dimension and the connectivity of M, we show that further information can be extracted by studying the structure of the fundamental group of M and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a d-dimensional manifold (d ⩾ 3) whose fundamental group is not free has at least 3d + 1 vertices. As a corollary, every d-dimensional homology sphere that admits a combinatorial triangulation with less than 3d vertices is PL-homeomorphic to Sd. Another important consequence is that every triangulation with small links of M is combinatorial.

Semi-equivelar maps on the torus and the Klein bottle with few vertices

Semi-equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the 2-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-equivelar maps on the torus and the Klein bottle with few vertices.

On a memory game and preferential attachment graphs

In their recent paper Velleman and Warrington (2013) analyzed the expected values of some of the parameters in a memory game; namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively. We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by-product of our methods, we obtain the joint asymptotic normality of the degree counts in the preferential attachment graphs. Furthermore, we obtain simpler proofs (although without rate of convergence) of some of the results of Peköz et al. (2014) on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs. In order to prove that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multitype generalized Pólya urn model.

The 0–1 inverse maximum independent set problem on forests and unicyclic graphs

Given a graph [Formula: see text] and an independent set [Formula: see text] of [Formula: see text], the 0–1 inverse maximum independent set problem (IMIS[Formula: see text]) is to delete as few vertices as possible such that [Formula: see text] becomes a maximum independent set of [Formula: see text]. It is known that IMIS[Formula: see text] is NP-hard even when the given independent set has a bounded size. In this paper, we present linear-time algorithms for IMIS[Formula: see text] on forests and unicyclic graphs.