Related papers: Evolutionary Hamiltonian Graph Theory
Dirac's classical theorem asserts that, for $n \ge 3$, any $n$-vertex graph with minimum degree at least $n/2$ is Hamiltonian. Furthermore, if we additionally assume that such graphs are regular, then, by the breakthrough work of Csaba,…
Nonlinearity and non-Hermiticity, for example due to environmental gain-loss processes, are a common occurrence throughout numerous areas of science and lie at the root of many remarkable phenomena. For the latter, parity-time-reflection…
Adaptive landscape has been a fundamental concept in many branches of modern biology since Wright's first proposition in 1932. Meanwhile, the general existence of landscape remains controversial. The causes include the mixed uses of…
A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. The problem of finding edge-disjoint Hamiltonian cycles in a given regular graph has many applications in combinatorial optimization and…
A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a…
Consider a graph having quantum systems lying at each node. Suppose that the whole thing evolves in discrete time steps, according to a global, unitary causal operator. By causal we mean that information can only propagate at a bounded…
In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\delta\geq 2$ contains a cycle with at least $\min\{2\delta,n\}$…
A dynamical theory for the evolution of the genetic code is presented, which accounts for its universality and optimality. The central concept is that a variety of collective, but non-Darwinian, mechanisms likely to be present in early…
We consider the following NP-hard problem: in a weighted graph, find a minimum cost set of vertices whose removal leaves a graph in which no two cycles share an edge. We obtain a constant-factor approximation algorithm, based on the…
In the Planar 3-SAT problem, we are given a 3-SAT formula together with its incidence graph, which is planar, and are asked whether this formula is satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it has been used…
Population dynamics in systems composed of cyclically competing species has been of increasing interest recently. Here, we investigate a system with four or more species. Using mean field theory, we study in detail the trajectories in…
The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers Asratian and Khachatryan developed local criteria for the existence of Hamilton cycles…
Predicting and comparing algorithm performance on graph instances is challenging for multiple reasons. First, there is usually no standard set of instances to benchmark performance. Second, using existing graph generators results in a…
Can a graph specifying the pattern of connections of a dynamical network be reconstructed from statistical properties of a signal generated by such a system? In this model study, we present an evolutionary algorithm for reconstruction of…
We investigate species-rich mathematical models of ecosystems. While much of the existing literature focuses on the properties of equilibrium fixed points, persistent dynamics (e.g., limit cycles or chaos) have also been observed, both in…
Existing theoretical models of evolution focus on the relative fitness advantages of different mutants in a population while the dynamic behavior of the population size is mostly left unconsidered. We here present a generic stochastic model…
Scientists world-wide are putting together massive efforts to understand how the biodiversity that we see on Earth evolved from single-cell organisms at the origin of life and this diversification process is represented through the Tree of…
The discovery of new exoplanets makes us wonder where each new exoplanet stands along its way to develop life as we know it on Earth. Our Evo-SETI Theory is a mathematical way to face this problem. We describe cladistics and evolution by…
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…
Evolution produces complex and structured networks of interacting components in chemical, biological, and social systems. We describe a simple mathematical model for the evolution of an idealized chemical system to study how a network of…