Number of cliques in a graph and intersection number The Next CEO of Stack OverflowMaximal and Maximum CliquesSelection from cliques of a graph in polynomial timeExample of graph with specific $chi (G)$, $omega (G)$, $beta (G)$If a graph G is the union of $n$ cliques of size $n$ no two of which share more than one vertex, then $chi_f(G)=n$. $omega(G)=n$How to optimize the clique problem when permuting a known graph?Graph Theory Connectivity ProofLet $G$ be a graph and $omega$ be its clique numberProve that for all $nge1$, a complete graph of n vertices contains k-cliques for k $in 1,…,n$Graph theoretic name for minimal subgraph that connects to full graphCounting the number of undirected simple and connected graphs
What day is it again?
Find a path from s to t using as few red nodes as possible
Traveling with my 5 year old daughter (as the father) without the mother from Germany to Mexico
That's an odd coin - I wonder why
logical reads on global temp table, but not on session-level temp table
How do I secure a TV wall mount?
Does int main() need a declaration on C++?
Why does freezing point matter when picking cooler ice packs?
Find the majority element, which appears more than half the time
Finitely generated matrix groups whose eigenvalues are all algebraic
Creating a script with console commands
Is it reasonable to ask other researchers to send me their previous grant applications?
Direct Implications Between USA and UK in Event of No-Deal Brexit
Prodigo = pro + ago?
Can Sri Krishna be called 'a person'?
Car headlights in a world without electricity
How does a dynamic QR code work?
Which acid/base does a strong base/acid react when added to a buffer solution?
What difference does it make matching a word with/without a trailing whitespace?
Upgrading From a 9 Speed Sora Derailleur?
Is a linearly independent set whose span is dense a Schauder basis?
Another proof that dividing by 0 does not exist -- is it right?
Is the offspring between a demon and a celestial possible? If so what is it called and is it in a book somewhere?
How can I separate the number from the unit in argument?
Number of cliques in a graph and intersection number
The Next CEO of Stack OverflowMaximal and Maximum CliquesSelection from cliques of a graph in polynomial timeExample of graph with specific $chi (G)$, $omega (G)$, $beta (G)$If a graph G is the union of $n$ cliques of size $n$ no two of which share more than one vertex, then $chi_f(G)=n$. $omega(G)=n$How to optimize the clique problem when permuting a known graph?Graph Theory Connectivity ProofLet $G$ be a graph and $omega$ be its clique numberProve that for all $nge1$, a complete graph of n vertices contains k-cliques for k $in 1,…,n$Graph theoretic name for minimal subgraph that connects to full graphCounting the number of undirected simple and connected graphs
$begingroup$
Define the number of cliques in a graph $G$ to be $c(G)$ and the intersection number of the graph to be $omega(G)$. I have been tasked to comment on the inequality between $c(G)$ and $omega(G)$. I believe that $c(G)geq omega(G)$. Consider $S_v$ to be the set corresponding to the vertex $v$. Define $C_i=v lvert iin S_v$ for each $iin S$. Now $C_i$ is a complete subgraph of $G$ and
also if $C_i=C_j$ then we can remove $j$ and everything would remain same, thus getting intersection number less than $omega(G)$ which is impossible.
I think the proof is correct, but I am a little unsure if I can indeed remove $j$, but to complete my task I must give an example where $c(G)>omega(G)$, I can't seem to find it. Some help would be appreciated. Thanks.
graph-theory
asked Feb 14 '16 at 14:57
shadow10shadow10
2,877931
$endgroup$
add a comment |
$begingroup$
Define the number of cliques in a graph $G$ to be $c(G)$ and the intersection number of the graph to be $omega(G)$. I have been tasked to comment on the inequality between $c(G)$ and $omega(G)$. I believe that $c(G)geq omega(G)$. Consider $S_v$ to be the set corresponding to the vertex $v$. Define $C_i=v lvert iin S_v$ for each $iin S$. Now $C_i$ is a complete subgraph of $G$ and
also if $C_i=C_j$ then we can remove $j$ and everything would remain same, thus getting intersection number less than $omega(G)$ which is impossible.
I think the proof is correct, but I am a little unsure if I can indeed remove $j$, but to complete my task I must give an example where $c(G)>omega(G)$, I can't seem to find it. Some help would be appreciated. Thanks.
graph-theory
asked Feb 14 '16 at 14:57
shadow10shadow10
2,877931
$endgroup$
$begingroup$
did you ever find an answer to this?
$endgroup$
– rachelhoward
Apr 26 '18 at 23:22
add a comment |
$begingroup$
Define the number of cliques in a graph $G$ to be $c(G)$ and the intersection number of the graph to be $omega(G)$. I have been tasked to comment on the inequality between $c(G)$ and $omega(G)$. I believe that $c(G)geq omega(G)$. Consider $S_v$ to be the set corresponding to the vertex $v$. Define $C_i=v lvert iin S_v$ for each $iin S$. Now $C_i$ is a complete subgraph of $G$ and
also if $C_i=C_j$ then we can remove $j$ and everything would remain same, thus getting intersection number less than $omega(G)$ which is impossible.
I think the proof is correct, but I am a little unsure if I can indeed remove $j$, but to complete my task I must give an example where $c(G)>omega(G)$, I can't seem to find it. Some help would be appreciated. Thanks.
graph-theory
asked Feb 14 '16 at 14:57
shadow10shadow10
2,877931
$endgroup$
Define the number of cliques in a graph $G$ to be $c(G)$ and the intersection number of the graph to be $omega(G)$. I have been tasked to comment on the inequality between $c(G)$ and $omega(G)$. I believe that $c(G)geq omega(G)$. Consider $S_v$ to be the set corresponding to the vertex $v$. Define $C_i=v lvert iin S_v$ for each $iin S$. Now $C_i$ is a complete subgraph of $G$ and
also if $C_i=C_j$ then we can remove $j$ and everything would remain same, thus getting intersection number less than $omega(G)$ which is impossible.
I think the proof is correct, but I am a little unsure if I can indeed remove $j$, but to complete my task I must give an example where $c(G)>omega(G)$, I can't seem to find it. Some help would be appreciated. Thanks.
graph-theory
graph-theory
asked Feb 14 '16 at 14:57
shadow10shadow10
2,877931
asked Feb 14 '16 at 14:57
shadow10shadow10
2,877931
asked Feb 14 '16 at 14:57
shadow10shadow10
2,877931
asked Feb 14 '16 at 14:57
shadow10shadow10
2,877931
asked Feb 14 '16 at 14:57
shadow10shadow10
2,877931
2,877931
$begingroup$
did you ever find an answer to this?
$endgroup$
– rachelhoward
Apr 26 '18 at 23:22
add a comment |
$begingroup$
did you ever find an answer to this?
$endgroup$
– rachelhoward
Apr 26 '18 at 23:22
$begingroup$
did you ever find an answer to this?
$endgroup$
– rachelhoward
Apr 26 '18 at 23:22
$begingroup$
did you ever find an answer to this?
$endgroup$
– rachelhoward
Apr 26 '18 at 23:22
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It is quite straightforward from the definition. You are comparing the number of cliques, with the minimum number of cliques that have a certain condition (that is, they cover the set of edges of the graph). In other words, you are comparing the size of a set, with the size of its subset.
It is known that the number of cliques in a graph is no smaller than the number of edges, i.e. $c(G) geq m$. Also, we know that the number of edges is an upper bound for the intersection number, i.e. $omega(G)leq m$. Doesn't this imply the proof that you are looking for?
answered Mar 28 at 9:56
orezvaniorezvani
254310
$endgroup$
add a comment |
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%2f1654500%2fnumber-of-cliques-in-a-graph-and-intersection-number%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
$begingroup$
It is quite straightforward from the definition. You are comparing the number of cliques, with the minimum number of cliques that have a certain condition (that is, they cover the set of edges of the graph). In other words, you are comparing the size of a set, with the size of its subset.
It is known that the number of cliques in a graph is no smaller than the number of edges, i.e. $c(G) geq m$. Also, we know that the number of edges is an upper bound for the intersection number, i.e. $omega(G)leq m$. Doesn't this imply the proof that you are looking for?
answered Mar 28 at 9:56
orezvaniorezvani
254310
$endgroup$
add a comment |
$begingroup$
It is quite straightforward from the definition. You are comparing the number of cliques, with the minimum number of cliques that have a certain condition (that is, they cover the set of edges of the graph). In other words, you are comparing the size of a set, with the size of its subset.
It is known that the number of cliques in a graph is no smaller than the number of edges, i.e. $c(G) geq m$. Also, we know that the number of edges is an upper bound for the intersection number, i.e. $omega(G)leq m$. Doesn't this imply the proof that you are looking for?
answered Mar 28 at 9:56
orezvaniorezvani
254310
$endgroup$
add a comment |
$begingroup$
It is quite straightforward from the definition. You are comparing the number of cliques, with the minimum number of cliques that have a certain condition (that is, they cover the set of edges of the graph). In other words, you are comparing the size of a set, with the size of its subset.
It is known that the number of cliques in a graph is no smaller than the number of edges, i.e. $c(G) geq m$. Also, we know that the number of edges is an upper bound for the intersection number, i.e. $omega(G)leq m$. Doesn't this imply the proof that you are looking for?
answered Mar 28 at 9:56
orezvaniorezvani
254310
$endgroup$
It is quite straightforward from the definition. You are comparing the number of cliques, with the minimum number of cliques that have a certain condition (that is, they cover the set of edges of the graph). In other words, you are comparing the size of a set, with the size of its subset.
It is known that the number of cliques in a graph is no smaller than the number of edges, i.e. $c(G) geq m$. Also, we know that the number of edges is an upper bound for the intersection number, i.e. $omega(G)leq m$. Doesn't this imply the proof that you are looking for?
answered Mar 28 at 9:56
orezvaniorezvani
254310
answered Mar 28 at 9:56
orezvaniorezvani
254310
answered Mar 28 at 9:56
orezvaniorezvani
254310
answered Mar 28 at 9:56
orezvaniorezvani
254310
254310
add a comment |
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%2f1654500%2fnumber-of-cliques-in-a-graph-and-intersection-number%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
$begingroup$
did you ever find an answer to this?
$endgroup$
– rachelhoward
Apr 26 '18 at 23:22