Hamiltonian graph


by Solarmew
Tags: graph, hamiltonian
Solarmew
Solarmew is offline
#1
Dec27-12, 04:33 PM
P: 36
Suppose G is a HC (Hamiltonian-connected) graph on n >= 4 vertices. Show that connectivity of G is 3.
I tried starting by saying that there would be at least 4C2=6 unique hamiltonian paths. But then I'm not sure where to go from here.
Any hints would be appreciated.
Phys.Org News Partner Science news on Phys.org
Going nuts? Turkey looks to pistachios to heat new eco-city
Space-tested fluid flow concept advances infectious disease diagnoses
SpaceX launches supplies to space station (Update)
haruspex
haruspex is online now
#2
Dec28-12, 10:15 PM
Homework
Sci Advisor
HW Helper
Thanks ∞
P: 9,160
I assume this is edge connectivity.
Suppose there's a cutset of two edges. In relation to these edges, find two points for which there can only be a Hamiltonian path between them under very restrictive conditions.
Solarmew
Solarmew is offline
#3
Dec28-12, 10:33 PM
P: 36
Quote Quote by haruspex View Post
I assume this is edge connectivity.
Suppose there's a cutset of two edges. In relation to these edges, find two points for which there can only be a Hamiltonian path between them under very restrictive conditions.
it's vertex connectivity, sorry >.> so the smallest number of vertices in any vertex cut of G is 3

haruspex
haruspex is online now
#4
Dec28-12, 10:40 PM
Homework
Sci Advisor
HW Helper
Thanks ∞
P: 9,160

Hamiltonian graph


Quote Quote by Solarmew View Post
it's vertex connectivity, sorry >.> so the smallest number of vertices in any vertex cut of G is 3
OK, same deal, only easier. You're trying to prove connectivity >= 3. So assume false. That means there's a pair vertices whose removal would leave a disconnected graph. Is there a Hamiltonian path between them?
Solarmew
Solarmew is offline
#5
Dec28-12, 10:58 PM
P: 36
Quote Quote by haruspex View Post
OK, same deal, only easier. You're trying to prove connectivity >= 3. So assume false. That means there's a pair vertices whose removal would leave a disconnected graph. Is there a Hamiltonian path between them?
There is. Since G is HC, there's a Hamiltonian Path b/w every two pairs of vertices.
But i'm not sure how there being a path is helpful >.>
hm, lemme think about that for a sec...
haruspex
haruspex is online now
#6
Dec28-12, 11:19 PM
Homework
Sci Advisor
HW Helper
Thanks ∞
P: 9,160
Quote Quote by Solarmew View Post
There is. Since G is HC, there's a Hamiltonian Path b/w every two pairs of vertices.
But i'm not sure how there being a path is helpful >.>
hm, lemme think about that for a sec...
No, I mean take any graph that has a pair of vertices whose removal would render the graph disconnected. Draw a diagram. How could it have a Hamiltonian path between those two vertices?
Solarmew
Solarmew is offline
#7
Dec28-12, 11:21 PM
P: 36
ooooh, there isn't a HP b/w them, i lied ... i was just testing ya XD .... by doodling it out i can kinda see it i think +.+ but i'm not sure how to put it in words ...
like if by removing those two vertices (let's say u,v) we disconnected the graph, that would mean that the only paths from one component to the other were through u and v. But that would mean that if we tried to find a Hamiltonian path from u to v we would not be able to go from one block of G to the other without passing through v, so we would only be able to go through the vertices of one block of G before inevitably ending up at v.
gah ... that doesn't sound very convincing >.< ... i need to work on my proof speak :<
haruspex
haruspex is online now
#8
Dec29-12, 02:52 PM
Homework
Sci Advisor
HW Helper
Thanks ∞
P: 9,160
Yes, that's the argument.
Solarmew
Solarmew is offline
#9
Dec29-12, 08:46 PM
P: 36
Quote Quote by haruspex View Post
Yes, that's the argument.
= ^.^ = thanks, i appreciate your help!


Register to reply

Related Discussions
How to convert this time-dependent Hamiltonian to time-independent Hamiltonian? Quantum Physics 0
Commutator of the Hamiltonian with Position and Hamiltonian with Momentum Advanced Physics Homework 2
Drawing the angular velocity graph from the acceleration vs. time graph? Introductory Physics Homework 5
how to do velocity-time graph to distance graph/acceleration to velocity graph? Classical Physics 1
manipulating inverse square graph into straight line graph Introductory Physics Homework 4