$(A,B)$ with $v in A$ $degree(v)$ odd, and any two vertices in A share even number of neighbours. Show there...
Let $G = ((A,B),E)$ be a bipartite graph in which each vertex of $A$ is of odd degree. Suppose that any two vertices of $A$ have an even number of common neighbours. Show that $G$ has a matching saturating $A$.
My attempt: We try to apply Hall's theorem and use induction on number of vertices in A.
Base case: There are two vertices in A. Then they share at least 2 vertices so there is a matching covering $A$ by Hall's theorem.
Inductive step, Suppose all bipartite graph $(A,B)$ with $|A|=n$ with the stated properties has a matching covering A. Now consider a bipartite graph $(A,B)$ such that $|A|=n+1$. Pick a vertex $v in A$ and consider $(A',B)=(A-{v},B)$ The resulting graph still satisfy the above condition hence has a matching covering A. Hence by Hall's theorem , $S subset A'$ satisfy $|N(S)|geq|S|$. When added ${v}$ back, then I'm not sure if $|N(S cup {v})| geq |S|+1$. So what's should I do next?
proof-verification graph-theory
asked Nov 24 at 16:18
mathnoob
1,786422
add a comment |
Let $G = ((A,B),E)$ be a bipartite graph in which each vertex of $A$ is of odd degree. Suppose that any two vertices of $A$ have an even number of common neighbours. Show that $G$ has a matching saturating $A$.
My attempt: We try to apply Hall's theorem and use induction on number of vertices in A.
Base case: There are two vertices in A. Then they share at least 2 vertices so there is a matching covering $A$ by Hall's theorem.
Inductive step, Suppose all bipartite graph $(A,B)$ with $|A|=n$ with the stated properties has a matching covering A. Now consider a bipartite graph $(A,B)$ such that $|A|=n+1$. Pick a vertex $v in A$ and consider $(A',B)=(A-{v},B)$ The resulting graph still satisfy the above condition hence has a matching covering A. Hence by Hall's theorem , $S subset A'$ satisfy $|N(S)|geq|S|$. When added ${v}$ back, then I'm not sure if $|N(S cup {v})| geq |S|+1$. So what's should I do next?
proof-verification graph-theory
asked Nov 24 at 16:18
mathnoob
1,786422
Possible duplicate of Matching in bipartite graphs
– Misha Lavrov
Nov 24 at 20:49
add a comment |
Let $G = ((A,B),E)$ be a bipartite graph in which each vertex of $A$ is of odd degree. Suppose that any two vertices of $A$ have an even number of common neighbours. Show that $G$ has a matching saturating $A$.
My attempt: We try to apply Hall's theorem and use induction on number of vertices in A.
Base case: There are two vertices in A. Then they share at least 2 vertices so there is a matching covering $A$ by Hall's theorem.
Inductive step, Suppose all bipartite graph $(A,B)$ with $|A|=n$ with the stated properties has a matching covering A. Now consider a bipartite graph $(A,B)$ such that $|A|=n+1$. Pick a vertex $v in A$ and consider $(A',B)=(A-{v},B)$ The resulting graph still satisfy the above condition hence has a matching covering A. Hence by Hall's theorem , $S subset A'$ satisfy $|N(S)|geq|S|$. When added ${v}$ back, then I'm not sure if $|N(S cup {v})| geq |S|+1$. So what's should I do next?
proof-verification graph-theory
asked Nov 24 at 16:18
mathnoob
1,786422
Let $G = ((A,B),E)$ be a bipartite graph in which each vertex of $A$ is of odd degree. Suppose that any two vertices of $A$ have an even number of common neighbours. Show that $G$ has a matching saturating $A$.
My attempt: We try to apply Hall's theorem and use induction on number of vertices in A.
Base case: There are two vertices in A. Then they share at least 2 vertices so there is a matching covering $A$ by Hall's theorem.
Inductive step, Suppose all bipartite graph $(A,B)$ with $|A|=n$ with the stated properties has a matching covering A. Now consider a bipartite graph $(A,B)$ such that $|A|=n+1$. Pick a vertex $v in A$ and consider $(A',B)=(A-{v},B)$ The resulting graph still satisfy the above condition hence has a matching covering A. Hence by Hall's theorem , $S subset A'$ satisfy $|N(S)|geq|S|$. When added ${v}$ back, then I'm not sure if $|N(S cup {v})| geq |S|+1$. So what's should I do next?
proof-verification graph-theory
proof-verification graph-theory
asked Nov 24 at 16:18
mathnoob
1,786422
asked Nov 24 at 16:18
mathnoob
1,786422
asked Nov 24 at 16:18
mathnoob
1,786422
asked Nov 24 at 16:18
mathnoob
1,786422
asked Nov 24 at 16:18
mathnoob
1,786422
1,786422
Possible duplicate of Matching in bipartite graphs
– Misha Lavrov
Nov 24 at 20:49
add a comment |
Possible duplicate of Matching in bipartite graphs
– Misha Lavrov
Nov 24 at 20:49
Possible duplicate of Matching in bipartite graphs
– Misha Lavrov
Nov 24 at 20:49
Possible duplicate of Matching in bipartite graphs
– Misha Lavrov
Nov 24 at 20:49
add a comment |
active
oldest
votes
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%2f3011741%2fa-b-with-v-in-a-degreev-odd-and-any-two-vertices-in-a-share-even-num%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3011741%2fa-b-with-v-in-a-degreev-odd-and-any-two-vertices-in-a-share-even-num%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
Possible duplicate of Matching in bipartite graphs
– Misha Lavrov
Nov 24 at 20:49