Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.

Hamiltonian Cycle Embeddings in Faulty Hypercubes Under the Forbidden Faulty Set Model

In this paper, we study the fault-tolerant capability of hypercubes with respect to the hamiltonian property based on the concept of forbidden faulty sets. We show, with the assumption that each vertex is incident with at least three fault-free edges, that an [Formula: see text]-dimensional hypercube contains a fault-free hamiltonian cycle, even if there are up to [Formula: see text] edge faults. Moreover, we give an example to show that the result is optimal with respect to the number of edge faults tolerated.

LIOUVILLE INTEGRABLE REDUCTIONS OF THE ASSOCIATIVITY EQUATIONS ON THE SET OF STATIONARY POINTS OF AN INTEGRAL IN THE CASE OF THREE PRIMARY FIELDS

In this work, in the case of three primary fields, a reduction of the associativity equations (the Witten–Dijkgraaf–Verlinde–Verlinde system, see (Witten, 1990, Dijkgraaf et al., 1991, Dubrovin, 1994) with antidiagonal matrix ηij on the set of stationary points of a nondegenerate integral quadratic with respect to the first-order partial derivatives is constructed in an explicit form and its Liouville integrability is proved. In Mokhov’s paper (Mokhov, 1995, Mokhov, 1998), these associativity equations were presented in the form of an integrable nondiagonalizable system of hydrodynamic type. In the papers (Ferapontov, Mokhov, 1996, Ferapontov et al., 1997, Mokhov, 1998), a bi-Hamiltonian representation for these equations and a nondegenerate integral quadratic with respect to the first-order partial derivatives were found. Using Mokhov’s construction on canonical Hamiltonian property of an arbitrary evolutionary system on the set of stationary points of its nondegenerate integral of the papers (Mokhov, 1984, Mokhov, 1987), we construct explicitly the reduction for the integral quadratic with respect to the first-order partial derivatives, found explicitly the Hamiltonian of the corresponding canonical Hamiltonian system. We also found three functionally-independent integrals in involution with respect to the canonical Poisson bracket on the phase space for the constructed reduction of the associativity equations and thus proved the Liouville integrability of this reduction. This work is supported by the Russian Science Foundation under grant No. 18-11-00316.

ON THE HAMILTONIAN REDUCTION OF THE ASSOCIATIVITY EQUATIONS IN THE CASE OF FOUR PRIMARY FIELDS

The associativity equations arose in the papers of Witten (Witten, 1990) and Dijkgraaf, Verlinde, (Dijkgraaf et al., 1991) on two-dimensional topological field theories and subsequently they became to play a key role in many other important domains of mathematics and mathematical physics: in quantum cohomology, Gromov–Witten invariants, enumerative geometry, theory of submanifolds and so on. In Mokhov’s papers (Mokhov, 1984), (Mokhov, 1987) a general fundamental principle stating a canonical Hamiltonian property for the restriction of an arbitrary flow on the set of stationary points of its nondegenerate integral was proposed and proved. In this paper the Hamiltonians of the reductions of the associativity equations with antidiagonal matrix ηij in the case of four primary fields according to Mokhov`s construction is found in an explicit form.

Path (or cycle)-trees with Graph Equations involving Line and Split Graphs

H-trees generalizes the existing notions of trees, higher dimensional trees and k-ctrees. The characterizations and properties of both Pk-trees for k at least 4 and Cn-trees for n at least 5 and their hamiltonian property, dominations, planarity, chromatic and b-chromatic numbers are established. The conditions under which Pk-trees for k at least 3 (resp. Cn-trees for n at least 4), are the line graphs are determined. The relationship between path-trees and split graphs are developed.