What is the maximum connectivity of a planar graph?
$begingroup$
The Icosahedral Graph is a simple 5-connected planar graph. Is there a 6-connected planar graph?
In general, is there a theoretical maximum on the vertex connectivity of planar graphs? This is spurred since the number of edges is bounded by $3n-6$, and that could set an upper bound on the vertex connectivity.
graph-theory algorithms connectedness graph-connectivity
asked Dec 4 '18 at 22:53
illigradilligrad
334
$endgroup$
add a comment |
$begingroup$
The Icosahedral Graph is a simple 5-connected planar graph. Is there a 6-connected planar graph?
In general, is there a theoretical maximum on the vertex connectivity of planar graphs? This is spurred since the number of edges is bounded by $3n-6$, and that could set an upper bound on the vertex connectivity.
graph-theory algorithms connectedness graph-connectivity
asked Dec 4 '18 at 22:53
illigradilligrad
334
$endgroup$
2
$begingroup$
A planar graph must have a vertex of degree less than six.
$endgroup$
– Gerry Myerson
Dec 4 '18 at 22:57
add a comment |
$begingroup$
The Icosahedral Graph is a simple 5-connected planar graph. Is there a 6-connected planar graph?
In general, is there a theoretical maximum on the vertex connectivity of planar graphs? This is spurred since the number of edges is bounded by $3n-6$, and that could set an upper bound on the vertex connectivity.
graph-theory algorithms connectedness graph-connectivity
asked Dec 4 '18 at 22:53
illigradilligrad
334
$endgroup$
The Icosahedral Graph is a simple 5-connected planar graph. Is there a 6-connected planar graph?
In general, is there a theoretical maximum on the vertex connectivity of planar graphs? This is spurred since the number of edges is bounded by $3n-6$, and that could set an upper bound on the vertex connectivity.
graph-theory algorithms connectedness graph-connectivity
graph-theory algorithms connectedness graph-connectivity
asked Dec 4 '18 at 22:53
illigradilligrad
334
asked Dec 4 '18 at 22:53
illigradilligrad
334
asked Dec 4 '18 at 22:53
illigradilligrad
334
asked Dec 4 '18 at 22:53
illigradilligrad
334
asked Dec 4 '18 at 22:53
illigradilligrad
334
334
2
$begingroup$
A planar graph must have a vertex of degree less than six.
$endgroup$
– Gerry Myerson
Dec 4 '18 at 22:57
add a comment |
2
$begingroup$
A planar graph must have a vertex of degree less than six.
$endgroup$
– Gerry Myerson
Dec 4 '18 at 22:57
2
2
$begingroup$
A planar graph must have a vertex of degree less than six.
$endgroup$
– Gerry Myerson
Dec 4 '18 at 22:57
$begingroup$
A planar graph must have a vertex of degree less than six.
$endgroup$
– Gerry Myerson
Dec 4 '18 at 22:57
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I think you pointed at the exact right direction.
As the number of edges is at most $3n-6$, there must be a vertex $v$ of degree at most $5$. Delete the neighbors of $v$, and the graph falls apart (unless it had at most 6 vertices).
answered Dec 4 '18 at 22:57
A. PongráczA. Pongrácz
5,9531929
$endgroup$
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%2f3026294%2fwhat-is-the-maximum-connectivity-of-a-planar-graph%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
$begingroup$
I think you pointed at the exact right direction.
As the number of edges is at most $3n-6$, there must be a vertex $v$ of degree at most $5$. Delete the neighbors of $v$, and the graph falls apart (unless it had at most 6 vertices).
answered Dec 4 '18 at 22:57
A. PongráczA. Pongrácz
5,9531929
$endgroup$
add a comment |
$begingroup$
I think you pointed at the exact right direction.
As the number of edges is at most $3n-6$, there must be a vertex $v$ of degree at most $5$. Delete the neighbors of $v$, and the graph falls apart (unless it had at most 6 vertices).
answered Dec 4 '18 at 22:57
A. PongráczA. Pongrácz
5,9531929
$endgroup$
add a comment |
$begingroup$
I think you pointed at the exact right direction.
As the number of edges is at most $3n-6$, there must be a vertex $v$ of degree at most $5$. Delete the neighbors of $v$, and the graph falls apart (unless it had at most 6 vertices).
answered Dec 4 '18 at 22:57
A. PongráczA. Pongrácz
5,9531929
$endgroup$
I think you pointed at the exact right direction.
As the number of edges is at most $3n-6$, there must be a vertex $v$ of degree at most $5$. Delete the neighbors of $v$, and the graph falls apart (unless it had at most 6 vertices).
answered Dec 4 '18 at 22:57
A. PongráczA. Pongrácz
5,9531929
edited Dec 4 '18 at 23:02
answered Dec 4 '18 at 22:57
A. PongráczA. Pongrácz
5,9531929
answered Dec 4 '18 at 22:57
A. PongráczA. Pongrácz
5,9531929
answered Dec 4 '18 at 22:57
A. PongráczA. Pongrácz
5,9531929
5,9531929
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%2f3026294%2fwhat-is-the-maximum-connectivity-of-a-planar-graph%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
2
$begingroup$
A planar graph must have a vertex of degree less than six.
$endgroup$
– Gerry Myerson
Dec 4 '18 at 22:57