Sharp thresholds for nonlinear Hamiltonian cycles in hypergraphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

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Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

Boundary Value Problems ◽

10.1186/s13661-019-01310-6 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Yongbin Wang ◽

Binhua Feng

Keyword(s):

Global Existence ◽

Finite Time ◽

Critical Case ◽

Blow Up ◽

Critical Mass ◽

Choquard Equation ◽

Power Type ◽

Scale Invariant ◽

Nonlinear Schrödinger ◽

Sharp Thresholds

AbstractIn this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$L2-critical case. Our obtained results extend and improve some recent results.

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Hamiltonian Cycles in Strong Products of Graphs

Canadian Mathematical Bulletin ◽

10.4153/cmb-1979-037-0 ◽

1979 ◽

Vol 22 (3) ◽

pp. 305-309 ◽

Cited By ~ 5

Author(s):

J. C. Bermond ◽

A. Germa ◽

M. C. Heydemann

Keyword(s):

Connected Graph ◽

Hamiltonian Cycles ◽

Strong Product

Abstract. Let denote the graph (k times) where is the strong product of the two graphs G and H. In this paper we prove the conjecture of J. Zaks [3]: For every connected graph G with at least two vertices there exists an integer k = k(G) for which the graph is hamiltonian.

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