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Every maximal outerplanar graph satisfies a stronger condition than Hamiltonicity: it is node pancyclic, meaning that for every vertex v and every k in the range from three to the number of vertices in the graph, there is a length-k cycle containing v. A cycle of this length may be found by repeatedly removing a triangle that is connected to the rest of the graph by a single edge, such that the removed vertex is not v, until the outer face of the remaining graph has length k.

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## Some Related Sentences

Every and maximal

** Zorn's lemma

**:**__Every__non-empty partially ordered set**in**which**every**chain ( i**.**e**.**totally ordered subset )**has**an upper bound contains at least one__maximal__element**.**
The restricted principle "

__Every__partially ordered set**has****a**__maximal__totally ordered subset "**is**also equivalent**to**AC over ZF**.**
** Tukey's lemma

**:**__Every__non-empty collection**of**finite character**has****a**__maximal__element with respect**to**inclusion**.**__Every__character

**is**automatically continuous

**from**

**A**

**to**C

**,**since

**the**kernel

**of**

**a**character

**is**

**a**

__maximal__ideal

**,**which

**is**closed

**.**

* In any ring R

**,****a**__maximal__ideal**is**an ideal M**that****is**__maximal__**in****the**set**of**all proper ideals**of**R**,**i**.**e**.**M**is**contained**in**exactly 2 ideals**of**R**,**namely M itself**and****the**entire ring R**.**__Every____maximal__ideal**is****in**fact prime**.**__Every__simple R-module

**is**isomorphic

**to**

**a**quotient R / m where m

**is**

**a**

__maximal__right ideal

**of**R

**.**By

**the**above paragraph

**,**any quotient R / m

**is**

**a**simple module

**.**

__Every__prime ideal P

**in**

**a**Boolean ring R

**is**

__maximal__

**:**

**the**quotient ring R / P

**is**an integral domain

**and**also

**a**Boolean ring

**,**so

**it**

**is**isomorphic

**to**

**the**field F < sub > 2 </ sub >, which shows

**the**maximality

**of**P

**.**Since

__maximal__ideals are always prime

**,**prime ideals

**and**

__maximal__ideals coincide

**in**Boolean rings

**.**

*

__Every__non-empty set**of**left ideals**of**R**,**partially ordered**by**inclusion**,****has****a**__maximal__element with respect**to**set inclusion**.**__Every__

__maximal__

**outerplanar**

**graph**with n

**vertices**

**has**exactly 2n − 3 edges

**,**

**and**

**every**bounded

**face**

**of**

**a**

__maximal__

**outerplanar**

**graph**

**is**

**a**

**triangle**

**.**

We call

**a**field E**a**splitting field**for****A**if**A**⊗ E**is**isomorphic**to****a**matrix ring over E**.**__Every__finite dimensional CSA**has****a**splitting field**:**indeed**,****in****the**case when**A****is****a**division algebra**,**then**a**__maximal__subfield**of****A****is****a**splitting field**.**__Every__

**graph**contains at most 3 < sup > n / 3 </ sup >

__maximal__independent sets

**,**but many graphs have far fewer

**.**

Every and outerplanar

__Every__

__outerplanar__

**graph**can

**be**represented as an intersection

**graph**

**of**axis-aligned rectangles

**in**

**the**plane

**,**so

__outerplanar__graphs have boxicity at most two

**.**

Every and graph

__Every__

**connected**

__graph__

**is**an expander ; however

**,**different

**connected**graphs have different expansion parameters

**.**

:"[...]

__Every__invariant**and**co-variant thus becomes expressible**by****a**__graph__precisely identical with**a**Kekuléan diagram or chemicograph**.**
*

__Every__**connected**__graph__G admits**a**spanning tree**,**which**is****a**tree**that**contains**every****vertex****of**G**and**whose edges are edges**of**G**.**
In mathematics

**,**Tait's conjecture states**that**"__Every__3-connected planar cubic__graph__**has****a**Hamiltonian**cycle**( along**the**edges ) through all its**vertices**".__Every__homomorphism

**of**

**the**Petersen

__graph__

**to**itself

**that**doesn't identify adjacent

**vertices**

**is**an automorphism

**.**

__Every__directed acyclic

__graph__

**has**

**a**topological ordering

**,**an ordering

**of**

**the**

**vertices**

**such**

**that**

**the**starting endpoint

**of**

**every**

**edge**occurs earlier

**in**

**the**ordering

**than**

**the**ending endpoint

**of**

**the**

**edge**

**.**

__Every__Eulerian orientation

**of**

**a**

**connected**

__graph__

**is**

**a**strong orientation

**,**an orientation

**that**makes

**the**resulting directed

__graph__strongly

**connected**

**.**

Every and satisfies

*

__Every__continuous functor on**a**small-complete category which__satisfies__**the**appropriate solution set**condition****has****a**left-adjoint (**the**Freyd adjoint functor theorem ).
#

__Every__finitely generated ideal**of****A****is**principal ( i**.**e.,**A****is****a**Bézout domain )**and****A**__satisfies__**the**ascending chain**condition**on principal ideals**.**__Every__bounded positive-definite measure μ on G

__satisfies__μ ( 1 ) ≥ 0

**.**improved

**this**criterion

**by**showing

**that**

**it**

**is**sufficient

**to**ask

**that**

**,**

**for**

**every**continuous positive-definite compactly supported function f on G

**,**

**the**function Δ < sup >– ½ </ sup > f

**has**non-negative integral with respect

**to**Haar measure

**,**where Δ denotes

**the**modular function

**.**

* Let K < sup >

**a**</ sup >**be**an algebraic closure**of**K**containing**L**.**__Every__embedding σ**of**L**in**K < sup >**a**</ sup > which restricts**to****the**identity on K**,**__satisfies__σ ( L ) = L**.**In other words**,**σ**is**an automorphism**of**L over K**.**0.748 seconds.