Graph Theory: Properties of even graph

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












The question is that: show that each even graph can be decomposed into edge-disjoint cycles.



What I think is: when drawing such graphs (randomly), I can produce two sets X and Y with vertices, in which XUY is all the vertice and X ∩Y is empty. But to prove in a mathematical way, how should be explained in words??






graph-theory







share|cite|improve this question





















  • Please see this tutorial and reference on how to typeset math on this site.
    – joriki
    Sep 6 at 9:29










  • Related: math.stackexchange.com/questions/2542753/…
    – Ivan Neretin
    Sep 6 at 9:32










  • by the way, is my understanding towards "edge-disjoint cycle" correct?
    – Jason Ng
    Sep 6 at 13:49










  • I still don't understand that answers... I am frustrated on my desk =_+
    – Jason Ng
    Sep 6 at 14:37










  • What is your definition of even graph? There is more than one way to understand it.
    – Michal Adamaszek
    Sep 7 at 15:38














up vote
0
down vote

favorite












The question is that: show that each even graph can be decomposed into edge-disjoint cycles.



What I think is: when drawing such graphs (randomly), I can produce two sets X and Y with vertices, in which XUY is all the vertice and X ∩Y is empty. But to prove in a mathematical way, how should be explained in words??






graph-theory







share|cite|improve this question





















  • Please see this tutorial and reference on how to typeset math on this site.
    – joriki
    Sep 6 at 9:29










  • Related: math.stackexchange.com/questions/2542753/…
    – Ivan Neretin
    Sep 6 at 9:32










  • by the way, is my understanding towards "edge-disjoint cycle" correct?
    – Jason Ng
    Sep 6 at 13:49










  • I still don't understand that answers... I am frustrated on my desk =_+
    – Jason Ng
    Sep 6 at 14:37










  • What is your definition of even graph? There is more than one way to understand it.
    – Michal Adamaszek
    Sep 7 at 15:38












up vote
0
down vote

favorite









up vote
0
down vote

favorite











The question is that: show that each even graph can be decomposed into edge-disjoint cycles.



What I think is: when drawing such graphs (randomly), I can produce two sets X and Y with vertices, in which XUY is all the vertice and X ∩Y is empty. But to prove in a mathematical way, how should be explained in words??






graph-theory







share|cite|improve this question













The question is that: show that each even graph can be decomposed into edge-disjoint cycles.



What I think is: when drawing such graphs (randomly), I can produce two sets X and Y with vertices, in which XUY is all the vertice and X ∩Y is empty. But to prove in a mathematical way, how should be explained in words??






graph-theory




graph-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Sep 6 at 9:07









Jason Ng

466




466











  • Please see this tutorial and reference on how to typeset math on this site.
    – joriki
    Sep 6 at 9:29










  • Related: math.stackexchange.com/questions/2542753/…
    – Ivan Neretin
    Sep 6 at 9:32










  • by the way, is my understanding towards "edge-disjoint cycle" correct?
    – Jason Ng
    Sep 6 at 13:49










  • I still don't understand that answers... I am frustrated on my desk =_+
    – Jason Ng
    Sep 6 at 14:37










  • What is your definition of even graph? There is more than one way to understand it.
    – Michal Adamaszek
    Sep 7 at 15:38
















  • Please see this tutorial and reference on how to typeset math on this site.
    – joriki
    Sep 6 at 9:29










  • Related: math.stackexchange.com/questions/2542753/…
    – Ivan Neretin
    Sep 6 at 9:32










  • by the way, is my understanding towards "edge-disjoint cycle" correct?
    – Jason Ng
    Sep 6 at 13:49










  • I still don't understand that answers... I am frustrated on my desk =_+
    – Jason Ng
    Sep 6 at 14:37










  • What is your definition of even graph? There is more than one way to understand it.
    – Michal Adamaszek
    Sep 7 at 15:38















Please see this tutorial and reference on how to typeset math on this site.
– joriki
Sep 6 at 9:29




Please see this tutorial and reference on how to typeset math on this site.
– joriki
Sep 6 at 9:29












Related: math.stackexchange.com/questions/2542753/…
– Ivan Neretin
Sep 6 at 9:32




Related: math.stackexchange.com/questions/2542753/…
– Ivan Neretin
Sep 6 at 9:32












by the way, is my understanding towards "edge-disjoint cycle" correct?
– Jason Ng
Sep 6 at 13:49




by the way, is my understanding towards "edge-disjoint cycle" correct?
– Jason Ng
Sep 6 at 13:49












I still don't understand that answers... I am frustrated on my desk =_+
– Jason Ng
Sep 6 at 14:37




I still don't understand that answers... I am frustrated on my desk =_+
– Jason Ng
Sep 6 at 14:37












What is your definition of even graph? There is more than one way to understand it.
– Michal Adamaszek
Sep 7 at 15:38




What is your definition of even graph? There is more than one way to understand it.
– Michal Adamaszek
Sep 7 at 15:38










1 Answer
1






active

oldest

votes

















up vote
0
down vote













If by even graph you mean all vertices have even degrees then you do as follows: start at any vertex and keep on walking, until you hit a vertex you already visited. That means you have a cycle. Remove the edges of that cycle from the graph. The remaining graph is still even. Proceed by induction.



The only thing you must notice is that "keep on walking" is well-defined, i.e. you won't get stuck, but that follows from the fact that the degree of each vertex is at least $2$. So if you entered a vertex, you can always find (some other) exit.






share|cite|improve this answer




















    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',
    convertImagesToLinks: true,
    noModals: false,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );













     

    draft saved


    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2907258%2fgraph-theory-properties-of-even-graph%23new-answer', 'question_page');

    );

    Post as a guest

















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    0
    down vote













    If by even graph you mean all vertices have even degrees then you do as follows: start at any vertex and keep on walking, until you hit a vertex you already visited. That means you have a cycle. Remove the edges of that cycle from the graph. The remaining graph is still even. Proceed by induction.



    The only thing you must notice is that "keep on walking" is well-defined, i.e. you won't get stuck, but that follows from the fact that the degree of each vertex is at least $2$. So if you entered a vertex, you can always find (some other) exit.






    share|cite|improve this answer
























      up vote
      0
      down vote













      If by even graph you mean all vertices have even degrees then you do as follows: start at any vertex and keep on walking, until you hit a vertex you already visited. That means you have a cycle. Remove the edges of that cycle from the graph. The remaining graph is still even. Proceed by induction.



      The only thing you must notice is that "keep on walking" is well-defined, i.e. you won't get stuck, but that follows from the fact that the degree of each vertex is at least $2$. So if you entered a vertex, you can always find (some other) exit.






      share|cite|improve this answer






















        up vote
        0
        down vote










        up vote
        0
        down vote









        If by even graph you mean all vertices have even degrees then you do as follows: start at any vertex and keep on walking, until you hit a vertex you already visited. That means you have a cycle. Remove the edges of that cycle from the graph. The remaining graph is still even. Proceed by induction.



        The only thing you must notice is that "keep on walking" is well-defined, i.e. you won't get stuck, but that follows from the fact that the degree of each vertex is at least $2$. So if you entered a vertex, you can always find (some other) exit.






        share|cite|improve this answer












        If by even graph you mean all vertices have even degrees then you do as follows: start at any vertex and keep on walking, until you hit a vertex you already visited. That means you have a cycle. Remove the edges of that cycle from the graph. The remaining graph is still even. Proceed by induction.



        The only thing you must notice is that "keep on walking" is well-defined, i.e. you won't get stuck, but that follows from the fact that the degree of each vertex is at least $2$. So if you entered a vertex, you can always find (some other) exit.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Sep 8 at 18:23









        Michal Adamaszek

        1,75448




        1,75448



























             

            draft saved


            draft discarded















































             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2907258%2fgraph-theory-properties-of-even-graph%23new-answer', 'question_page');

            );

            Post as a guest
































































            -

            這個網誌中的熱門文章

            How to combine Bézier curves to a surface?

            圖片
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...
            閱讀完整內容

            Mutual Information Always Non-negative

            圖片
            Clash Royale CLAN TAG #URR8PPP up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 '12 at 15:38 Cameron Buie 83.6k 7 71 154 asked Jun 17 '12 at 15:30 Omri 359 3 13 3 Convexity of the function $tmapsto tlog t$. – Did Jun 17 '12 at 15:33 In addition, the convexity properties require the coefficients in the linear combination sum 1. Then, as p(x,y) is a probability distribution, it fullfits such condition. – Francisco Javier Delgado Ceped Aug 10 at 18:44 add a comment  |  up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 ...
            閱讀完整內容

            Why am i infinitely getting the same tweet with the Twitter Search API?

            圖片
            up vote 0 down vote favorite My code workes perfectly for other queries, but for one query I am infinitely getting the same tweet. I can't understand what the problem is. API call: query='Allergic asthma OR Nonallergic asthma OR Occupational asthma OR EIB OR Exercise-induced bronchoconstriction OR Nocturnal asthma OR Cough-variant asthma' new_tweets = api.search(q=searchQuery, count=100, since_id=since_id, lang='en',tweet_mode='extended') if not new_tweets: print("No more tweets found") break for tweet in new_tweets: if True: print(jsonpickle.encode(tweet._json, unpicklable=False)) my_file = open('searchTweets.txt','a') my_file.write(jsonpickle.encode(tweet._json, unpicklable=False)+'n') my_file.close() else: continue Output in file: https://pastebin.com/JmRCsTeE These are the JSONs of the same Tweet. Other queries that work: Breast cancer OR Prostate cancer OR Colon cancer OR Lung cancer Type 2 dia...
            閱讀完整內容