Are G and H necessarily isomorphic?












0














Let $G$ and $H$ be two simple graphs, both of them with seven vertices, each of which is of degree 2. Are G and H necessarily isomorphic?



The graphs $G$ and $H$ must have a cycle since each vertex is of degree 2 and therefore they are isomorphic.In general, this should be true for a graph with $n$ vertices with the above property? Is this correct?










share|cite|improve this question



























    0














    Let $G$ and $H$ be two simple graphs, both of them with seven vertices, each of which is of degree 2. Are G and H necessarily isomorphic?



    The graphs $G$ and $H$ must have a cycle since each vertex is of degree 2 and therefore they are isomorphic.In general, this should be true for a graph with $n$ vertices with the above property? Is this correct?










    share|cite|improve this question

























      0












      0








      0







      Let $G$ and $H$ be two simple graphs, both of them with seven vertices, each of which is of degree 2. Are G and H necessarily isomorphic?



      The graphs $G$ and $H$ must have a cycle since each vertex is of degree 2 and therefore they are isomorphic.In general, this should be true for a graph with $n$ vertices with the above property? Is this correct?










      share|cite|improve this question













      Let $G$ and $H$ be two simple graphs, both of them with seven vertices, each of which is of degree 2. Are G and H necessarily isomorphic?



      The graphs $G$ and $H$ must have a cycle since each vertex is of degree 2 and therefore they are isomorphic.In general, this should be true for a graph with $n$ vertices with the above property? Is this correct?







      graph-theory graph-isomorphism






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 2 at 16:40









      thetraveller

      1515




      1515






















          1 Answer
          1






          active

          oldest

          votes


















          0














          No; one of them could be $C_7$ (the $7$-vertex cycle) and the other could be the disjoint union of $C_3$ and $C_4$.



          If you additionally knew that the graphs were connected, then there would be only one possibility.






          share|cite|improve this answer





















          • You're right, it didn't say connected...should pay attention to definition. But if they were then the answer is correct right?
            – thetraveller
            Dec 2 at 16:54










          • Yes, if they were connected, then they would both have to be $C_7$. (And in the general $n$-vertex case, $C_n$ is the only connected $2$-regular graph.)
            – Misha Lavrov
            Dec 2 at 16:54













          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


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3022861%2fare-g-and-h-necessarily-isomorphic%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









          0














          No; one of them could be $C_7$ (the $7$-vertex cycle) and the other could be the disjoint union of $C_3$ and $C_4$.



          If you additionally knew that the graphs were connected, then there would be only one possibility.






          share|cite|improve this answer





















          • You're right, it didn't say connected...should pay attention to definition. But if they were then the answer is correct right?
            – thetraveller
            Dec 2 at 16:54










          • Yes, if they were connected, then they would both have to be $C_7$. (And in the general $n$-vertex case, $C_n$ is the only connected $2$-regular graph.)
            – Misha Lavrov
            Dec 2 at 16:54


















          0














          No; one of them could be $C_7$ (the $7$-vertex cycle) and the other could be the disjoint union of $C_3$ and $C_4$.



          If you additionally knew that the graphs were connected, then there would be only one possibility.






          share|cite|improve this answer





















          • You're right, it didn't say connected...should pay attention to definition. But if they were then the answer is correct right?
            – thetraveller
            Dec 2 at 16:54










          • Yes, if they were connected, then they would both have to be $C_7$. (And in the general $n$-vertex case, $C_n$ is the only connected $2$-regular graph.)
            – Misha Lavrov
            Dec 2 at 16:54
















          0












          0








          0






          No; one of them could be $C_7$ (the $7$-vertex cycle) and the other could be the disjoint union of $C_3$ and $C_4$.



          If you additionally knew that the graphs were connected, then there would be only one possibility.






          share|cite|improve this answer












          No; one of them could be $C_7$ (the $7$-vertex cycle) and the other could be the disjoint union of $C_3$ and $C_4$.



          If you additionally knew that the graphs were connected, then there would be only one possibility.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 2 at 16:45









          Misha Lavrov

          43.8k555104




          43.8k555104












          • You're right, it didn't say connected...should pay attention to definition. But if they were then the answer is correct right?
            – thetraveller
            Dec 2 at 16:54










          • Yes, if they were connected, then they would both have to be $C_7$. (And in the general $n$-vertex case, $C_n$ is the only connected $2$-regular graph.)
            – Misha Lavrov
            Dec 2 at 16:54




















          • You're right, it didn't say connected...should pay attention to definition. But if they were then the answer is correct right?
            – thetraveller
            Dec 2 at 16:54










          • Yes, if they were connected, then they would both have to be $C_7$. (And in the general $n$-vertex case, $C_n$ is the only connected $2$-regular graph.)
            – Misha Lavrov
            Dec 2 at 16:54


















          You're right, it didn't say connected...should pay attention to definition. But if they were then the answer is correct right?
          – thetraveller
          Dec 2 at 16:54




          You're right, it didn't say connected...should pay attention to definition. But if they were then the answer is correct right?
          – thetraveller
          Dec 2 at 16:54












          Yes, if they were connected, then they would both have to be $C_7$. (And in the general $n$-vertex case, $C_n$ is the only connected $2$-regular graph.)
          – Misha Lavrov
          Dec 2 at 16:54






          Yes, if they were connected, then they would both have to be $C_7$. (And in the general $n$-vertex case, $C_n$ is the only connected $2$-regular graph.)
          – Misha Lavrov
          Dec 2 at 16:54




















          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.





          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.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3022861%2fare-g-and-h-necessarily-isomorphic%23new-answer', 'question_page');
          }
          );

          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







          Popular posts from this blog

          Berounka

          Sphinx de Gizeh

          Different font size/position of beamer's navigation symbols template's content depending on regular/plain...