This paper is a contribution to the presentation of fractal sets in terms of final coal-gebras. The first result on this topic was Freyd's Theorem: the unit interval [0, 1] is the final coalgebra of a functor X → X ⊕ X on the category of bipointed sets. Leinster [L] offers a sweeping generalization of this result. He is able to represent many of what would be intuitively called self-similar spaces using (a) bimodules (also called profunctors or distributors), (b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction of final coalgebras for the types of functors of interest using a notion of resolution. In addition to the characterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces. Our major contribution is to suggest that in many cases of interest, point (c) above on resolutions is not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces of interest as the Cauchy completion of an initial algebra, and this initial algebra is the set of points in a colimit of an ω-sequence of finite metric spaces. This generalizes Hutchinson's characterization of fractal attractors in [H] as closures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is "computationally related" to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space of dyadic rational numbers in [0, 1]. Our second contribution is not completed at this time, but it is a set of results on metric space characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui [HJN], and our interest in quotient metrics comes from [HJN]. So in terms of (a)-(c) above, our work develops (a) and (b) in metric settings while dropping (c).

This paper is a contribution to the presentation of fractal sets in terms of final coalgebras. The first result on this topic was Freyd's Theorem: the unit interval [0, 1] is the final coalgebra of a functor X → X ⊕ X on the category of bipointed sets. Leinster [L] offers a sweeping generalization of this result. He is able to represent many of what would be intuitively called self-similar spaces using (a) bimodules (also called profunctors or distributors), (b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction of final coalgebras for the types of functors of interest using a notion of resolution. In addition to the characterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces.Our major contribution is to suggest that in many cases of interest, point (c) above on resolutions is not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces of interest as the Cauchy completion of an initial algebra, and this initial algebra is the set of points in a colimit of an ω-sequence of finite metric spaces. This generalizes Hutchinson's characterization of fractal attractors in [H] as closures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is "computationally related" to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space of dyadic rational numbers in [0, 1]. Our second contribution is not completed at this time, but it is a set of results on metric space characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui [HJN], and our interest in quotient metrics comes from [HJN]. So in terms of (a)-(c) above, our work develops (a) and (b) in metric settings while dropping (c).

Lovász had posed a question stating whether every connected, vertex-transitive graph has a Hamilton path in 1969. There is a growing interest in solving this longstanding problem and still it remains widely open. In fact, it was known that only five vertex-transitive graphs exist without a Hamiltonian cycle which do not belong to Cayley graphs. A Cayley graph is the subclass of vertex-transitive graph, and in view of the Lovász conjecture, the attention has focused more toward the Hamiltonicity of Cayley graphs. This survey will describe the current status of the search for Hamiltonian cycles and paths in Cayley graphs and digraphs on different groups, and discuss the future direction regarding famous conjecture.

Given j ≥ 2, for graphs G and H, the size Ramsey multipartite number m j (G, H) is defined as the smallest natural number t such that any blue red coloring of the edges of the graph K j×t , necessarily containes a red G or a blue H as subgraphs. Let the book with n pages is defined as the graph K 1 + K 1,n and denoted by B n . In this paper, we obtain the exact values of the size Ramsey numbers m j (P 3 , H) for j ≥ 3 where H is a book B n . We also derive some upper and lower bounds for the size Ramsey numbers m j (P 4 , H) where H is a book B n .

The metric space of p-adic integers with the p-adic distinguished point, we lift Z p Cauchy completion of the initial algebra of an endofunctor on the category of one-pointed one-bounded metric spaces with distinguished point, we lift Z p is also maps are known to be the Cauchy completion of the intial distinguished point, we lift Z p coalgebra of certain endofunctors on ultra metric spaces. The results of this paper unify these observations and give a coalgebraic characterisation of the self similarity of distinguished point, we lift Z p relaxing the ultra metric condition to one-bounded metrics. the initial algebra is in close analogy with classical results in iterated function systems. Another question that has been asked in the literature is whether such results hold when the maps are continuous maps as the morphisms may be the right choice for such results to hold.