Prove or disprove: If graph $G$ has at most $3n-1$ edges, then $chi(G) leq 6$
0
$begingroup$
The question is to prove or disprove that if graph $G$ has at most $3n-1$ edges, then the chromatic number is at most $6$. I tried to work it out using the fact that $chi(G) leq triangle(G) + 1$, but I can't think of a relation between the max degree and the number of edges.
graph-theory
asked Dec 9 '18 at 14:27
Mera InsanMera Insan
176
$endgroup$
|
show 1 more comment
0
$begingroup$
The question is to prove or disprove that if graph $G$ has at most $3n-1$ edges, then the chromatic number is at most $6$. I tried to work it out using the fact that $chi(G) leq triangle(G) + 1$, but I can't think of a relation between the max degree and the number of edges.
graph-theory
asked Dec 9 '18 at 14:27
Mera InsanMera Insan
176
$endgroup$
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 9 '18 at 14:35
1
$begingroup$
This seems impossible: for every fixed $k$, consider the complete graph on $k$ vertices and append to it a long linear portion, then, if the number of vertices is $n$, when $ntoinfty$, the number of edges is $n+O(1)$ and the chromatic number is (at least) $k$. For your homework, take $k=7$.
$endgroup$
– Did
Dec 9 '18 at 14:37
$begingroup$
But maybe $n$ is not the number of vertices of $G$... What is $n$?
$endgroup$
– Did
Dec 9 '18 at 14:42
$begingroup$
n is the number of vertices of G. I don't get what you meant that the number of edges increases by 1 when n goes to infinity. Thanks for your help
$endgroup$
– Mera Insan
Dec 9 '18 at 14:47
$begingroup$
Or what did you mean by n+O(1)
$endgroup$
– Mera Insan
Dec 9 '18 at 14:49
|
show 1 more comment
0
0
0
$begingroup$
The question is to prove or disprove that if graph $G$ has at most $3n-1$ edges, then the chromatic number is at most $6$. I tried to work it out using the fact that $chi(G) leq triangle(G) + 1$, but I can't think of a relation between the max degree and the number of edges.
graph-theory
asked Dec 9 '18 at 14:27
Mera InsanMera Insan
176
$endgroup$
The question is to prove or disprove that if graph $G$ has at most $3n-1$ edges, then the chromatic number is at most $6$. I tried to work it out using the fact that $chi(G) leq triangle(G) + 1$, but I can't think of a relation between the max degree and the number of edges.
graph-theory
graph-theory
asked Dec 9 '18 at 14:27
Mera InsanMera Insan
176
asked Dec 9 '18 at 14:27
Mera InsanMera Insan
176
edited Dec 9 '18 at 14:46
Mera Insan
asked Dec 9 '18 at 14:27
Mera InsanMera Insan
176
asked Dec 9 '18 at 14:27
Mera InsanMera Insan
176
asked Dec 9 '18 at 14:27
Mera InsanMera Insan
176
176
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 9 '18 at 14:35
1
$begingroup$
This seems impossible: for every fixed $k$, consider the complete graph on $k$ vertices and append to it a long linear portion, then, if the number of vertices is $n$, when $ntoinfty$, the number of edges is $n+O(1)$ and the chromatic number is (at least) $k$. For your homework, take $k=7$.
$endgroup$
– Did
Dec 9 '18 at 14:37
$begingroup$
But maybe $n$ is not the number of vertices of $G$... What is $n$?
$endgroup$
– Did
Dec 9 '18 at 14:42
$begingroup$
n is the number of vertices of G. I don't get what you meant that the number of edges increases by 1 when n goes to infinity. Thanks for your help
$endgroup$
– Mera Insan
Dec 9 '18 at 14:47
$begingroup$
Or what did you mean by n+O(1)
$endgroup$
– Mera Insan
Dec 9 '18 at 14:49
|
show 1 more comment
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 9 '18 at 14:35
1
$begingroup$
This seems impossible: for every fixed $k$, consider the complete graph on $k$ vertices and append to it a long linear portion, then, if the number of vertices is $n$, when $ntoinfty$, the number of edges is $n+O(1)$ and the chromatic number is (at least) $k$. For your homework, take $k=7$.
$endgroup$
– Did
Dec 9 '18 at 14:37
$begingroup$
But maybe $n$ is not the number of vertices of $G$... What is $n$?
$endgroup$
– Did
Dec 9 '18 at 14:42
$begingroup$
n is the number of vertices of G. I don't get what you meant that the number of edges increases by 1 when n goes to infinity. Thanks for your help
$endgroup$
– Mera Insan
Dec 9 '18 at 14:47
$begingroup$
Or what did you mean by n+O(1)
$endgroup$
– Mera Insan
Dec 9 '18 at 14:49
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 9 '18 at 14:35
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 9 '18 at 14:35
1
1
$begingroup$
This seems impossible: for every fixed $k$, consider the complete graph on $k$ vertices and append to it a long linear portion, then, if the number of vertices is $n$, when $ntoinfty$, the number of edges is $n+O(1)$ and the chromatic number is (at least) $k$. For your homework, take $k=7$.
$endgroup$
– Did
Dec 9 '18 at 14:37
$begingroup$
This seems impossible: for every fixed $k$, consider the complete graph on $k$ vertices and append to it a long linear portion, then, if the number of vertices is $n$, when $ntoinfty$, the number of edges is $n+O(1)$ and the chromatic number is (at least) $k$. For your homework, take $k=7$.
$endgroup$
– Did
Dec 9 '18 at 14:37
$begingroup$
But maybe $n$ is not the number of vertices of $G$... What is $n$?
$endgroup$
– Did
Dec 9 '18 at 14:42
$begingroup$
But maybe $n$ is not the number of vertices of $G$... What is $n$?
$endgroup$
– Did
Dec 9 '18 at 14:42
$begingroup$
n is the number of vertices of G. I don't get what you meant that the number of edges increases by 1 when n goes to infinity. Thanks for your help
$endgroup$
– Mera Insan
Dec 9 '18 at 14:47
$begingroup$
n is the number of vertices of G. I don't get what you meant that the number of edges increases by 1 when n goes to infinity. Thanks for your help
$endgroup$
– Mera Insan
Dec 9 '18 at 14:47
$begingroup$
Or what did you mean by n+O(1)
$endgroup$
– Mera Insan
Dec 9 '18 at 14:49
$begingroup$
Or what did you mean by n+O(1)
$endgroup$
– Mera Insan
Dec 9 '18 at 14:49
|
show 1 more comment
1 Answer
1
active
oldest
votes
1
$begingroup$
Ok so, consider $K_7$, It has $21$ edges which $3*7$. But then add a path of length $2$ to it to get the graph $G$ like in the following picture:
Then $chi(G)=7$, $n=9$and $E(G)=23 <3*n-1=26$. So dosent't work!
answered Dec 9 '18 at 14:58
nafhgoodnafhgood
1,797422
$endgroup$
$begingroup$
Thanks! I get it now
$endgroup$
– Mera Insan
Dec 9 '18 at 15:05
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
});
}
});
draft saved
draft discarded
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%2f3032422%2fprove-or-disprove-if-graph-g-has-at-most-3n-1-edges-then-chig-leq-6%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
1
$begingroup$
Ok so, consider $K_7$, It has $21$ edges which $3*7$. But then add a path of length $2$ to it to get the graph $G$ like in the following picture:
Then $chi(G)=7$, $n=9$and $E(G)=23 <3*n-1=26$. So dosent't work!
answered Dec 9 '18 at 14:58
nafhgoodnafhgood
1,797422
$endgroup$
$begingroup$
Thanks! I get it now
$endgroup$
– Mera Insan
Dec 9 '18 at 15:05
add a comment |
1
$begingroup$
Ok so, consider $K_7$, It has $21$ edges which $3*7$. But then add a path of length $2$ to it to get the graph $G$ like in the following picture:
Then $chi(G)=7$, $n=9$and $E(G)=23 <3*n-1=26$. So dosent't work!
answered Dec 9 '18 at 14:58
nafhgoodnafhgood
1,797422
$endgroup$
$begingroup$
Thanks! I get it now
$endgroup$
– Mera Insan
Dec 9 '18 at 15:05
add a comment |
1
1
1
$begingroup$
Ok so, consider $K_7$, It has $21$ edges which $3*7$. But then add a path of length $2$ to it to get the graph $G$ like in the following picture:
Then $chi(G)=7$, $n=9$and $E(G)=23 <3*n-1=26$. So dosent't work!
answered Dec 9 '18 at 14:58
nafhgoodnafhgood
1,797422
$endgroup$
Ok so, consider $K_7$, It has $21$ edges which $3*7$. But then add a path of length $2$ to it to get the graph $G$ like in the following picture:
Then $chi(G)=7$, $n=9$and $E(G)=23 <3*n-1=26$. So dosent't work!
answered Dec 9 '18 at 14:58
nafhgoodnafhgood
1,797422
answered Dec 9 '18 at 14:58
nafhgoodnafhgood
1,797422
answered Dec 9 '18 at 14:58
nafhgoodnafhgood
1,797422
answered Dec 9 '18 at 14:58
nafhgoodnafhgood
1,797422
1,797422
$begingroup$
Thanks! I get it now
$endgroup$
– Mera Insan
Dec 9 '18 at 15:05
add a comment |
$begingroup$
Thanks! I get it now
$endgroup$
– Mera Insan
Dec 9 '18 at 15:05
$begingroup$
Thanks! I get it now
$endgroup$
– Mera Insan
Dec 9 '18 at 15:05
$begingroup$
Thanks! I get it now
$endgroup$
– Mera Insan
Dec 9 '18 at 15:05
add a comment |
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f3032422%2fprove-or-disprove-if-graph-g-has-at-most-3n-1-edges-then-chig-leq-6%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
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 9 '18 at 14:35
1
$begingroup$
This seems impossible: for every fixed $k$, consider the complete graph on $k$ vertices and append to it a long linear portion, then, if the number of vertices is $n$, when $ntoinfty$, the number of edges is $n+O(1)$ and the chromatic number is (at least) $k$. For your homework, take $k=7$.
$endgroup$
– Did
Dec 9 '18 at 14:37
$begingroup$
But maybe $n$ is not the number of vertices of $G$... What is $n$?
$endgroup$
– Did
Dec 9 '18 at 14:42
$begingroup$
n is the number of vertices of G. I don't get what you meant that the number of edges increases by 1 when n goes to infinity. Thanks for your help
$endgroup$
– Mera Insan
Dec 9 '18 at 14:47
$begingroup$
Or what did you mean by n+O(1)
$endgroup$
– Mera Insan
Dec 9 '18 at 14:49