Proof that if all vertices have degree at least two then $G$ contains a cycle
$begingroup$
Prove that if all vertices have degree at least two then $G$ contains a cycle.
Here is the proof, but please correct me if wrong :
We assume $G$ is simple and let $P$ be the longest path $=v_0v_1v_2ldots v_{a-1}v_a$.
As it is given that the degree of $v_a$ is even ,then $v_a$ must have $mathbf{2}$ neigbors; one of them is $v_{a-1}$ and the other must be any $v$ in the Graph $G$, so $v=v_i$, where $i$ is any integer between $0$ and $a-2$ : $0leq i leq(a-2)$ #
Please correct me if I am wrong, and sorry for any mistakes.
combinatorics discrete-mathematics graph-theory trees
$endgroup$
add a comment |
$begingroup$
Prove that if all vertices have degree at least two then $G$ contains a cycle.
Here is the proof, but please correct me if wrong :
We assume $G$ is simple and let $P$ be the longest path $=v_0v_1v_2ldots v_{a-1}v_a$.
As it is given that the degree of $v_a$ is even ,then $v_a$ must have $mathbf{2}$ neigbors; one of them is $v_{a-1}$ and the other must be any $v$ in the Graph $G$, so $v=v_i$, where $i$ is any integer between $0$ and $a-2$ : $0leq i leq(a-2)$ #
Please correct me if I am wrong, and sorry for any mistakes.
combinatorics discrete-mathematics graph-theory trees
$endgroup$
$begingroup$
You can omit the [incorrect] claim that the degree of $v_a$ is even... you already know it has at least two neighbors
$endgroup$
– angryavian
Jun 22 '15 at 17:32
$begingroup$
Put the whole subscript in curly braces: to get $v_{a-1}$ usev_{a-1}
.
$endgroup$
– Brian M. Scott
Jun 22 '15 at 17:37
add a comment |
$begingroup$
Prove that if all vertices have degree at least two then $G$ contains a cycle.
Here is the proof, but please correct me if wrong :
We assume $G$ is simple and let $P$ be the longest path $=v_0v_1v_2ldots v_{a-1}v_a$.
As it is given that the degree of $v_a$ is even ,then $v_a$ must have $mathbf{2}$ neigbors; one of them is $v_{a-1}$ and the other must be any $v$ in the Graph $G$, so $v=v_i$, where $i$ is any integer between $0$ and $a-2$ : $0leq i leq(a-2)$ #
Please correct me if I am wrong, and sorry for any mistakes.
combinatorics discrete-mathematics graph-theory trees
$endgroup$
Prove that if all vertices have degree at least two then $G$ contains a cycle.
Here is the proof, but please correct me if wrong :
We assume $G$ is simple and let $P$ be the longest path $=v_0v_1v_2ldots v_{a-1}v_a$.
As it is given that the degree of $v_a$ is even ,then $v_a$ must have $mathbf{2}$ neigbors; one of them is $v_{a-1}$ and the other must be any $v$ in the Graph $G$, so $v=v_i$, where $i$ is any integer between $0$ and $a-2$ : $0leq i leq(a-2)$ #
Please correct me if I am wrong, and sorry for any mistakes.
combinatorics discrete-mathematics graph-theory trees
combinatorics discrete-mathematics graph-theory trees
edited Dec 20 '18 at 2:05
Batominovski
33.2k33293
33.2k33293
asked Jun 22 '15 at 17:28
John KantiJohn Kanti
217
217
$begingroup$
You can omit the [incorrect] claim that the degree of $v_a$ is even... you already know it has at least two neighbors
$endgroup$
– angryavian
Jun 22 '15 at 17:32
$begingroup$
Put the whole subscript in curly braces: to get $v_{a-1}$ usev_{a-1}
.
$endgroup$
– Brian M. Scott
Jun 22 '15 at 17:37
add a comment |
$begingroup$
You can omit the [incorrect] claim that the degree of $v_a$ is even... you already know it has at least two neighbors
$endgroup$
– angryavian
Jun 22 '15 at 17:32
$begingroup$
Put the whole subscript in curly braces: to get $v_{a-1}$ usev_{a-1}
.
$endgroup$
– Brian M. Scott
Jun 22 '15 at 17:37
$begingroup$
You can omit the [incorrect] claim that the degree of $v_a$ is even... you already know it has at least two neighbors
$endgroup$
– angryavian
Jun 22 '15 at 17:32
$begingroup$
You can omit the [incorrect] claim that the degree of $v_a$ is even... you already know it has at least two neighbors
$endgroup$
– angryavian
Jun 22 '15 at 17:32
$begingroup$
Put the whole subscript in curly braces: to get $v_{a-1}$ use
v_{a-1}
.$endgroup$
– Brian M. Scott
Jun 22 '15 at 17:37
$begingroup$
Put the whole subscript in curly braces: to get $v_{a-1}$ use
v_{a-1}
.$endgroup$
– Brian M. Scott
Jun 22 '15 at 17:37
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The idea is right, but there’s a small mistake, and there are a couple of places where you could be a little clearer. You don’t know that the degree of $v_a$ is even: it’s quite possible that $deg v_a$ is odd. You do know, however, that $deg v_age 2$, which is all that you need. Then you can say that $v_a$ must be adjacent to at least one vertex $v$ that is not $v_{a-1}$. At that point you really should say a little more than you did: you should point out that if $vnotin{v_0,ldots,v_{a-2}}$, then we could extent the path $P$ to $v$. However, $P$ was chosen to be of maximal length, so this is impossible, and therefore $vin{v_0ldots,v_{a-2}}$. Thus, if $v=v_i$, then $v_iv_{i+1}ldots v_av_i$ is a cycle.
$endgroup$
$begingroup$
I just I assume it is a string of nodes <contradiction> at the end ? is not it ?
$endgroup$
– John Kanti
Jun 23 '15 at 10:46
$begingroup$
@John: I’m not sure what you mean. The only place where a contradiction is involved is when we observe that $v$ is one of the vertices $v_0,ldots,v_{n-2}$: if it were not, the path $v_0ldots v_nv$ would contradict the choice of $P$ as a maximal path. There’s no contradiction at the end, though: there we simply have the desired cycle.
$endgroup$
– Brian M. Scott
Jun 23 '15 at 19:36
add a comment |
$begingroup$
I guess you assume that $G$ is a finite graph. What is the maximum possible number of edges that a spanning forest of $G$ can have? What is the minimum possible number of edges can $G$ have?
$endgroup$
$begingroup$
if you know the definition of a Simple Graph then you do not need to ask about the Graph limitation ? why you need the maximum possible number ? suppose maximum is X or any +number
$endgroup$
– John Kanti
Jun 22 '15 at 17:35
2
$begingroup$
Let $n$ be the number of vertices of $G$. If $G$ has no cycle, then $G$ is a forest, then $G$ has at most $n-1$ edges. If every vertex of $G$ is of degree at least $2$, then $G$ has at least $n$ edges. Do you see my point now?
$endgroup$
– Batominovski
Jun 23 '15 at 3:56
$begingroup$
not clear, why you need # of nodes, the theory does not tell about how many nodes we need, it says it is true whatever the number of nodes is. but at least each node must have at least 2 degrees. Did you get it ?!
$endgroup$
– John Kanti
Jun 23 '15 at 10:49
1
$begingroup$
No, I don't get what you said. Actually, I think it is you who doesn't "get it!" Unless you can make a coherent counter-argument, I will stop my conversation with you.
$endgroup$
– Batominovski
Jun 23 '15 at 10:58
$begingroup$
please read the Theory carefully, It does not talk about the number of nodes? so why you assume that we need to know the number of nodes? it is not the key point, Stopping conversation is your decision , I do not care about it
$endgroup$
– John Kanti
Jun 23 '15 at 11:01
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1335183%2fproof-that-if-all-vertices-have-degree-at-least-two-then-g-contains-a-cycle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The idea is right, but there’s a small mistake, and there are a couple of places where you could be a little clearer. You don’t know that the degree of $v_a$ is even: it’s quite possible that $deg v_a$ is odd. You do know, however, that $deg v_age 2$, which is all that you need. Then you can say that $v_a$ must be adjacent to at least one vertex $v$ that is not $v_{a-1}$. At that point you really should say a little more than you did: you should point out that if $vnotin{v_0,ldots,v_{a-2}}$, then we could extent the path $P$ to $v$. However, $P$ was chosen to be of maximal length, so this is impossible, and therefore $vin{v_0ldots,v_{a-2}}$. Thus, if $v=v_i$, then $v_iv_{i+1}ldots v_av_i$ is a cycle.
$endgroup$
$begingroup$
I just I assume it is a string of nodes <contradiction> at the end ? is not it ?
$endgroup$
– John Kanti
Jun 23 '15 at 10:46
$begingroup$
@John: I’m not sure what you mean. The only place where a contradiction is involved is when we observe that $v$ is one of the vertices $v_0,ldots,v_{n-2}$: if it were not, the path $v_0ldots v_nv$ would contradict the choice of $P$ as a maximal path. There’s no contradiction at the end, though: there we simply have the desired cycle.
$endgroup$
– Brian M. Scott
Jun 23 '15 at 19:36
add a comment |
$begingroup$
The idea is right, but there’s a small mistake, and there are a couple of places where you could be a little clearer. You don’t know that the degree of $v_a$ is even: it’s quite possible that $deg v_a$ is odd. You do know, however, that $deg v_age 2$, which is all that you need. Then you can say that $v_a$ must be adjacent to at least one vertex $v$ that is not $v_{a-1}$. At that point you really should say a little more than you did: you should point out that if $vnotin{v_0,ldots,v_{a-2}}$, then we could extent the path $P$ to $v$. However, $P$ was chosen to be of maximal length, so this is impossible, and therefore $vin{v_0ldots,v_{a-2}}$. Thus, if $v=v_i$, then $v_iv_{i+1}ldots v_av_i$ is a cycle.
$endgroup$
$begingroup$
I just I assume it is a string of nodes <contradiction> at the end ? is not it ?
$endgroup$
– John Kanti
Jun 23 '15 at 10:46
$begingroup$
@John: I’m not sure what you mean. The only place where a contradiction is involved is when we observe that $v$ is one of the vertices $v_0,ldots,v_{n-2}$: if it were not, the path $v_0ldots v_nv$ would contradict the choice of $P$ as a maximal path. There’s no contradiction at the end, though: there we simply have the desired cycle.
$endgroup$
– Brian M. Scott
Jun 23 '15 at 19:36
add a comment |
$begingroup$
The idea is right, but there’s a small mistake, and there are a couple of places where you could be a little clearer. You don’t know that the degree of $v_a$ is even: it’s quite possible that $deg v_a$ is odd. You do know, however, that $deg v_age 2$, which is all that you need. Then you can say that $v_a$ must be adjacent to at least one vertex $v$ that is not $v_{a-1}$. At that point you really should say a little more than you did: you should point out that if $vnotin{v_0,ldots,v_{a-2}}$, then we could extent the path $P$ to $v$. However, $P$ was chosen to be of maximal length, so this is impossible, and therefore $vin{v_0ldots,v_{a-2}}$. Thus, if $v=v_i$, then $v_iv_{i+1}ldots v_av_i$ is a cycle.
$endgroup$
The idea is right, but there’s a small mistake, and there are a couple of places where you could be a little clearer. You don’t know that the degree of $v_a$ is even: it’s quite possible that $deg v_a$ is odd. You do know, however, that $deg v_age 2$, which is all that you need. Then you can say that $v_a$ must be adjacent to at least one vertex $v$ that is not $v_{a-1}$. At that point you really should say a little more than you did: you should point out that if $vnotin{v_0,ldots,v_{a-2}}$, then we could extent the path $P$ to $v$. However, $P$ was chosen to be of maximal length, so this is impossible, and therefore $vin{v_0ldots,v_{a-2}}$. Thus, if $v=v_i$, then $v_iv_{i+1}ldots v_av_i$ is a cycle.
answered Jun 22 '15 at 17:34
Brian M. ScottBrian M. Scott
460k40517918
460k40517918
$begingroup$
I just I assume it is a string of nodes <contradiction> at the end ? is not it ?
$endgroup$
– John Kanti
Jun 23 '15 at 10:46
$begingroup$
@John: I’m not sure what you mean. The only place where a contradiction is involved is when we observe that $v$ is one of the vertices $v_0,ldots,v_{n-2}$: if it were not, the path $v_0ldots v_nv$ would contradict the choice of $P$ as a maximal path. There’s no contradiction at the end, though: there we simply have the desired cycle.
$endgroup$
– Brian M. Scott
Jun 23 '15 at 19:36
add a comment |
$begingroup$
I just I assume it is a string of nodes <contradiction> at the end ? is not it ?
$endgroup$
– John Kanti
Jun 23 '15 at 10:46
$begingroup$
@John: I’m not sure what you mean. The only place where a contradiction is involved is when we observe that $v$ is one of the vertices $v_0,ldots,v_{n-2}$: if it were not, the path $v_0ldots v_nv$ would contradict the choice of $P$ as a maximal path. There’s no contradiction at the end, though: there we simply have the desired cycle.
$endgroup$
– Brian M. Scott
Jun 23 '15 at 19:36
$begingroup$
I just I assume it is a string of nodes <contradiction> at the end ? is not it ?
$endgroup$
– John Kanti
Jun 23 '15 at 10:46
$begingroup$
I just I assume it is a string of nodes <contradiction> at the end ? is not it ?
$endgroup$
– John Kanti
Jun 23 '15 at 10:46
$begingroup$
@John: I’m not sure what you mean. The only place where a contradiction is involved is when we observe that $v$ is one of the vertices $v_0,ldots,v_{n-2}$: if it were not, the path $v_0ldots v_nv$ would contradict the choice of $P$ as a maximal path. There’s no contradiction at the end, though: there we simply have the desired cycle.
$endgroup$
– Brian M. Scott
Jun 23 '15 at 19:36
$begingroup$
@John: I’m not sure what you mean. The only place where a contradiction is involved is when we observe that $v$ is one of the vertices $v_0,ldots,v_{n-2}$: if it were not, the path $v_0ldots v_nv$ would contradict the choice of $P$ as a maximal path. There’s no contradiction at the end, though: there we simply have the desired cycle.
$endgroup$
– Brian M. Scott
Jun 23 '15 at 19:36
add a comment |
$begingroup$
I guess you assume that $G$ is a finite graph. What is the maximum possible number of edges that a spanning forest of $G$ can have? What is the minimum possible number of edges can $G$ have?
$endgroup$
$begingroup$
if you know the definition of a Simple Graph then you do not need to ask about the Graph limitation ? why you need the maximum possible number ? suppose maximum is X or any +number
$endgroup$
– John Kanti
Jun 22 '15 at 17:35
2
$begingroup$
Let $n$ be the number of vertices of $G$. If $G$ has no cycle, then $G$ is a forest, then $G$ has at most $n-1$ edges. If every vertex of $G$ is of degree at least $2$, then $G$ has at least $n$ edges. Do you see my point now?
$endgroup$
– Batominovski
Jun 23 '15 at 3:56
$begingroup$
not clear, why you need # of nodes, the theory does not tell about how many nodes we need, it says it is true whatever the number of nodes is. but at least each node must have at least 2 degrees. Did you get it ?!
$endgroup$
– John Kanti
Jun 23 '15 at 10:49
1
$begingroup$
No, I don't get what you said. Actually, I think it is you who doesn't "get it!" Unless you can make a coherent counter-argument, I will stop my conversation with you.
$endgroup$
– Batominovski
Jun 23 '15 at 10:58
$begingroup$
please read the Theory carefully, It does not talk about the number of nodes? so why you assume that we need to know the number of nodes? it is not the key point, Stopping conversation is your decision , I do not care about it
$endgroup$
– John Kanti
Jun 23 '15 at 11:01
add a comment |
$begingroup$
I guess you assume that $G$ is a finite graph. What is the maximum possible number of edges that a spanning forest of $G$ can have? What is the minimum possible number of edges can $G$ have?
$endgroup$
$begingroup$
if you know the definition of a Simple Graph then you do not need to ask about the Graph limitation ? why you need the maximum possible number ? suppose maximum is X or any +number
$endgroup$
– John Kanti
Jun 22 '15 at 17:35
2
$begingroup$
Let $n$ be the number of vertices of $G$. If $G$ has no cycle, then $G$ is a forest, then $G$ has at most $n-1$ edges. If every vertex of $G$ is of degree at least $2$, then $G$ has at least $n$ edges. Do you see my point now?
$endgroup$
– Batominovski
Jun 23 '15 at 3:56
$begingroup$
not clear, why you need # of nodes, the theory does not tell about how many nodes we need, it says it is true whatever the number of nodes is. but at least each node must have at least 2 degrees. Did you get it ?!
$endgroup$
– John Kanti
Jun 23 '15 at 10:49
1
$begingroup$
No, I don't get what you said. Actually, I think it is you who doesn't "get it!" Unless you can make a coherent counter-argument, I will stop my conversation with you.
$endgroup$
– Batominovski
Jun 23 '15 at 10:58
$begingroup$
please read the Theory carefully, It does not talk about the number of nodes? so why you assume that we need to know the number of nodes? it is not the key point, Stopping conversation is your decision , I do not care about it
$endgroup$
– John Kanti
Jun 23 '15 at 11:01
add a comment |
$begingroup$
I guess you assume that $G$ is a finite graph. What is the maximum possible number of edges that a spanning forest of $G$ can have? What is the minimum possible number of edges can $G$ have?
$endgroup$
I guess you assume that $G$ is a finite graph. What is the maximum possible number of edges that a spanning forest of $G$ can have? What is the minimum possible number of edges can $G$ have?
answered Jun 22 '15 at 17:32
BatominovskiBatominovski
33.2k33293
33.2k33293
$begingroup$
if you know the definition of a Simple Graph then you do not need to ask about the Graph limitation ? why you need the maximum possible number ? suppose maximum is X or any +number
$endgroup$
– John Kanti
Jun 22 '15 at 17:35
2
$begingroup$
Let $n$ be the number of vertices of $G$. If $G$ has no cycle, then $G$ is a forest, then $G$ has at most $n-1$ edges. If every vertex of $G$ is of degree at least $2$, then $G$ has at least $n$ edges. Do you see my point now?
$endgroup$
– Batominovski
Jun 23 '15 at 3:56
$begingroup$
not clear, why you need # of nodes, the theory does not tell about how many nodes we need, it says it is true whatever the number of nodes is. but at least each node must have at least 2 degrees. Did you get it ?!
$endgroup$
– John Kanti
Jun 23 '15 at 10:49
1
$begingroup$
No, I don't get what you said. Actually, I think it is you who doesn't "get it!" Unless you can make a coherent counter-argument, I will stop my conversation with you.
$endgroup$
– Batominovski
Jun 23 '15 at 10:58
$begingroup$
please read the Theory carefully, It does not talk about the number of nodes? so why you assume that we need to know the number of nodes? it is not the key point, Stopping conversation is your decision , I do not care about it
$endgroup$
– John Kanti
Jun 23 '15 at 11:01
add a comment |
$begingroup$
if you know the definition of a Simple Graph then you do not need to ask about the Graph limitation ? why you need the maximum possible number ? suppose maximum is X or any +number
$endgroup$
– John Kanti
Jun 22 '15 at 17:35
2
$begingroup$
Let $n$ be the number of vertices of $G$. If $G$ has no cycle, then $G$ is a forest, then $G$ has at most $n-1$ edges. If every vertex of $G$ is of degree at least $2$, then $G$ has at least $n$ edges. Do you see my point now?
$endgroup$
– Batominovski
Jun 23 '15 at 3:56
$begingroup$
not clear, why you need # of nodes, the theory does not tell about how many nodes we need, it says it is true whatever the number of nodes is. but at least each node must have at least 2 degrees. Did you get it ?!
$endgroup$
– John Kanti
Jun 23 '15 at 10:49
1
$begingroup$
No, I don't get what you said. Actually, I think it is you who doesn't "get it!" Unless you can make a coherent counter-argument, I will stop my conversation with you.
$endgroup$
– Batominovski
Jun 23 '15 at 10:58
$begingroup$
please read the Theory carefully, It does not talk about the number of nodes? so why you assume that we need to know the number of nodes? it is not the key point, Stopping conversation is your decision , I do not care about it
$endgroup$
– John Kanti
Jun 23 '15 at 11:01
$begingroup$
if you know the definition of a Simple Graph then you do not need to ask about the Graph limitation ? why you need the maximum possible number ? suppose maximum is X or any +number
$endgroup$
– John Kanti
Jun 22 '15 at 17:35
$begingroup$
if you know the definition of a Simple Graph then you do not need to ask about the Graph limitation ? why you need the maximum possible number ? suppose maximum is X or any +number
$endgroup$
– John Kanti
Jun 22 '15 at 17:35
2
2
$begingroup$
Let $n$ be the number of vertices of $G$. If $G$ has no cycle, then $G$ is a forest, then $G$ has at most $n-1$ edges. If every vertex of $G$ is of degree at least $2$, then $G$ has at least $n$ edges. Do you see my point now?
$endgroup$
– Batominovski
Jun 23 '15 at 3:56
$begingroup$
Let $n$ be the number of vertices of $G$. If $G$ has no cycle, then $G$ is a forest, then $G$ has at most $n-1$ edges. If every vertex of $G$ is of degree at least $2$, then $G$ has at least $n$ edges. Do you see my point now?
$endgroup$
– Batominovski
Jun 23 '15 at 3:56
$begingroup$
not clear, why you need # of nodes, the theory does not tell about how many nodes we need, it says it is true whatever the number of nodes is. but at least each node must have at least 2 degrees. Did you get it ?!
$endgroup$
– John Kanti
Jun 23 '15 at 10:49
$begingroup$
not clear, why you need # of nodes, the theory does not tell about how many nodes we need, it says it is true whatever the number of nodes is. but at least each node must have at least 2 degrees. Did you get it ?!
$endgroup$
– John Kanti
Jun 23 '15 at 10:49
1
1
$begingroup$
No, I don't get what you said. Actually, I think it is you who doesn't "get it!" Unless you can make a coherent counter-argument, I will stop my conversation with you.
$endgroup$
– Batominovski
Jun 23 '15 at 10:58
$begingroup$
No, I don't get what you said. Actually, I think it is you who doesn't "get it!" Unless you can make a coherent counter-argument, I will stop my conversation with you.
$endgroup$
– Batominovski
Jun 23 '15 at 10:58
$begingroup$
please read the Theory carefully, It does not talk about the number of nodes? so why you assume that we need to know the number of nodes? it is not the key point, Stopping conversation is your decision , I do not care about it
$endgroup$
– John Kanti
Jun 23 '15 at 11:01
$begingroup$
please read the Theory carefully, It does not talk about the number of nodes? so why you assume that we need to know the number of nodes? it is not the key point, Stopping conversation is your decision , I do not care about it
$endgroup$
– John Kanti
Jun 23 '15 at 11:01
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1335183%2fproof-that-if-all-vertices-have-degree-at-least-two-then-g-contains-a-cycle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
You can omit the [incorrect] claim that the degree of $v_a$ is even... you already know it has at least two neighbors
$endgroup$
– angryavian
Jun 22 '15 at 17:32
$begingroup$
Put the whole subscript in curly braces: to get $v_{a-1}$ use
v_{a-1}
.$endgroup$
– Brian M. Scott
Jun 22 '15 at 17:37