{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-26T00:14:02.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-26T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":45382,"title":"Find a Hamiltonian Cycle in a Graph","description":"You are given a graph g and asked to find a Hamiltonian cycle of g.\r\n\r\nSee \u003chttps://www.mathworks.com/help/matlab/ref/graph.html MATLAB graph documentation\u003e for details of the graph data structure.\r\n\r\nA cycle of g is a sequence of vertices of g such that each adjacent pair of vertices in the sequence share an edge in g and the first and last vertices in the sequence share an edge in g. A Hamiltonian cycle of g is a cycle of g that visits each vertex of g exactly once.\r\n\r\nFor example, consider the adjacency matrix below.\r\n\r\n  A = [\r\n     0     0     0     1     1     0     1     1\r\n     0     1     0     1     0     0     1     0\r\n     0     0     0     1     1     0     0     1\r\n     1     1     1     1     0     1     0     1\r\n     1     0     1     0     0     1     0     1\r\n     0     0     0     1     1     0     0     1\r\n     1     1     0     0     0     0     0     0\r\n     1     0     1     1     1     1     0     0];\r\n\r\nThis corresponds to the graph with vertices labeled 1 through 8 and an edge between two vertices i and j if and only if A(i, j) == 1. This graph has cycles of vertices [3 4 8], [3 8 1 4], and [5 1 4 3 8], among others. Try the commands below to visualize this.\r\n\r\n  g  = graph(A);\r\n  gh = plot(g);\r\n\r\nA Hamiltonian cycle for this graph g is [1 5 6 8 3 4 2 7].\r\n\r\nFor another fun challenge, see: \u003chttps://www.mathworks.com/matlabcentral/cody/problems/45252-restricted-addition-v1 Restricted Addition\u003e","description_html":"\u003cp\u003eYou are given a graph g and asked to find a Hamiltonian cycle of g.\u003c/p\u003e\u003cp\u003eSee \u003ca href = \"https://www.mathworks.com/help/matlab/ref/graph.html\"\u003eMATLAB graph documentation\u003c/a\u003e for details of the graph data structure.\u003c/p\u003e\u003cp\u003eA cycle of g is a sequence of vertices of g such that each adjacent pair of vertices in the sequence share an edge in g and the first and last vertices in the sequence share an edge in g. A Hamiltonian cycle of g is a cycle of g that visits each vertex of g exactly once.\u003c/p\u003e\u003cp\u003eFor example, consider the adjacency matrix below.\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eA = [\r\n   0     0     0     1     1     0     1     1\r\n   0     1     0     1     0     0     1     0\r\n   0     0     0     1     1     0     0     1\r\n   1     1     1     1     0     1     0     1\r\n   1     0     1     0     0     1     0     1\r\n   0     0     0     1     1     0     0     1\r\n   1     1     0     0     0     0     0     0\r\n   1     0     1     1     1     1     0     0];\r\n\u003c/pre\u003e\u003cp\u003eThis corresponds to the graph with vertices labeled 1 through 8 and an edge between two vertices i and j if and only if A(i, j) == 1. This graph has cycles of vertices [3 4 8], [3 8 1 4], and [5 1 4 3 8], among others. Try the commands below to visualize this.\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eg  = graph(A);\r\ngh = plot(g);\r\n\u003c/pre\u003e\u003cp\u003eA Hamiltonian cycle for this graph g is [1 5 6 8 3 4 2 7].\u003c/p\u003e\u003cp\u003eFor another fun challenge, see: \u003ca href = \"https://www.mathworks.com/matlabcentral/cody/problems/45252-restricted-addition-v1\"\u003eRestricted Addition\u003c/a\u003e\u003c/p\u003e","function_template":"function c = findCycle(g)\r\n    c = 1 : g.numnodes;\r\nend","test_suite":"%%\r\nA = [\r\n   0     0     0     1     1     0     1     1\r\n   0     1     0     1     0     0     1     0\r\n   0     0     0     1     1     0     0     1\r\n   1     1     1     1     0     1     0     1\r\n   1     0     1     0     0     1     0     1\r\n   0     0     0     1     1     0     0     1\r\n   1     1     0     0     0     0     0     0\r\n   1     0     1     1     1     1     0     0];\r\ng = graph(A);\r\nc = findCycle(g);\r\nn = 8;\r\nfor vI = 1 : n\r\n\tv = c(vI);\r\n\tu = c(mod(vI + 1, n) + n * (vI == n - 1));\r\n\t\r\n\tassert(findedge(g, v, u) \u003e 0);\r\nend\r\n\r\n%%\r\nn = 30;\r\np = 0.3;\r\nfor pI = 1 : 5\r\n\tn = n * 2;\r\n\tp = p / 2;\r\n\t\r\n\tx       = rand(n) \u003c p;\r\n\tx       = x + diag(ones(1, n - 1), -1);\r\n\tx(n, 1) = 1;\r\n\tx       = x + x.';\r\n\tr       = randperm(n);\r\n\tx(r, r) = x \u003e 0;\r\n\tg       = graph(x);\r\n\t\r\n\tc = findCycle(g);\r\n\tfor vI = 1 : n\r\n\t\tv = c(vI);\r\n\t\tu = c(mod(vI + 1, n) + n * (vI == n - 1));\r\n\t\t\r\n\t\tassert(findedge(g, v, u) \u003e 0);\r\n\tend\r\nend","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":692,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-03-24T02:36:00.000Z","updated_at":"2020-03-24T02:37:44.000Z","published_at":"2020-03-24T02:37:44.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou are given a graph g and asked to find a Hamiltonian cycle of g.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/matlab/ref/graph.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eMATLAB graph documentation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for details of the graph data structure.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA cycle of g is a sequence of vertices of g such that each adjacent pair of vertices in the sequence share an edge in g and the first and last vertices in the sequence share an edge in g. A Hamiltonian cycle of g is a cycle of g that visits each vertex of g exactly once.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, consider the adjacency matrix below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[A = [\\n   0     0     0     1     1     0     1     1\\n   0     1     0     1     0     0     1     0\\n   0     0     0     1     1     0     0     1\\n   1     1     1     1     0     1     0     1\\n   1     0     1     0     0     1     0     1\\n   0     0     0     1     1     0     0     1\\n   1     1     0     0     0     0     0     0\\n   1     0     1     1     1     1     0     0];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis corresponds to the graph with vertices labeled 1 through 8 and an edge between two vertices i and j if and only if A(i, j) == 1. This graph has cycles of vertices [3 4 8], [3 8 1 4], and [5 1 4 3 8], among others. Try the commands below to visualize this.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[g  = graph(A);\\ngh = plot(g);]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA Hamiltonian cycle for this graph g is [1 5 6 8 3 4 2 7].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor another fun challenge, see:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45252-restricted-addition-v1\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eRestricted Addition\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":45382,"title":"Find a Hamiltonian Cycle in a Graph","description":"You are given a graph g and asked to find a Hamiltonian cycle of g.\r\n\r\nSee \u003chttps://www.mathworks.com/help/matlab/ref/graph.html MATLAB graph documentation\u003e for details of the graph data structure.\r\n\r\nA cycle of g is a sequence of vertices of g such that each adjacent pair of vertices in the sequence share an edge in g and the first and last vertices in the sequence share an edge in g. A Hamiltonian cycle of g is a cycle of g that visits each vertex of g exactly once.\r\n\r\nFor example, consider the adjacency matrix below.\r\n\r\n  A = [\r\n     0     0     0     1     1     0     1     1\r\n     0     1     0     1     0     0     1     0\r\n     0     0     0     1     1     0     0     1\r\n     1     1     1     1     0     1     0     1\r\n     1     0     1     0     0     1     0     1\r\n     0     0     0     1     1     0     0     1\r\n     1     1     0     0     0     0     0     0\r\n     1     0     1     1     1     1     0     0];\r\n\r\nThis corresponds to the graph with vertices labeled 1 through 8 and an edge between two vertices i and j if and only if A(i, j) == 1. This graph has cycles of vertices [3 4 8], [3 8 1 4], and [5 1 4 3 8], among others. Try the commands below to visualize this.\r\n\r\n  g  = graph(A);\r\n  gh = plot(g);\r\n\r\nA Hamiltonian cycle for this graph g is [1 5 6 8 3 4 2 7].\r\n\r\nFor another fun challenge, see: \u003chttps://www.mathworks.com/matlabcentral/cody/problems/45252-restricted-addition-v1 Restricted Addition\u003e","description_html":"\u003cp\u003eYou are given a graph g and asked to find a Hamiltonian cycle of g.\u003c/p\u003e\u003cp\u003eSee \u003ca href = \"https://www.mathworks.com/help/matlab/ref/graph.html\"\u003eMATLAB graph documentation\u003c/a\u003e for details of the graph data structure.\u003c/p\u003e\u003cp\u003eA cycle of g is a sequence of vertices of g such that each adjacent pair of vertices in the sequence share an edge in g and the first and last vertices in the sequence share an edge in g. A Hamiltonian cycle of g is a cycle of g that visits each vertex of g exactly once.\u003c/p\u003e\u003cp\u003eFor example, consider the adjacency matrix below.\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eA = [\r\n   0     0     0     1     1     0     1     1\r\n   0     1     0     1     0     0     1     0\r\n   0     0     0     1     1     0     0     1\r\n   1     1     1     1     0     1     0     1\r\n   1     0     1     0     0     1     0     1\r\n   0     0     0     1     1     0     0     1\r\n   1     1     0     0     0     0     0     0\r\n   1     0     1     1     1     1     0     0];\r\n\u003c/pre\u003e\u003cp\u003eThis corresponds to the graph with vertices labeled 1 through 8 and an edge between two vertices i and j if and only if A(i, j) == 1. This graph has cycles of vertices [3 4 8], [3 8 1 4], and [5 1 4 3 8], among others. Try the commands below to visualize this.\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eg  = graph(A);\r\ngh = plot(g);\r\n\u003c/pre\u003e\u003cp\u003eA Hamiltonian cycle for this graph g is [1 5 6 8 3 4 2 7].\u003c/p\u003e\u003cp\u003eFor another fun challenge, see: \u003ca href = \"https://www.mathworks.com/matlabcentral/cody/problems/45252-restricted-addition-v1\"\u003eRestricted Addition\u003c/a\u003e\u003c/p\u003e","function_template":"function c = findCycle(g)\r\n    c = 1 : g.numnodes;\r\nend","test_suite":"%%\r\nA = [\r\n   0     0     0     1     1     0     1     1\r\n   0     1     0     1     0     0     1     0\r\n   0     0     0     1     1     0     0     1\r\n   1     1     1     1     0     1     0     1\r\n   1     0     1     0     0     1     0     1\r\n   0     0     0     1     1     0     0     1\r\n   1     1     0     0     0     0     0     0\r\n   1     0     1     1     1     1     0     0];\r\ng = graph(A);\r\nc = findCycle(g);\r\nn = 8;\r\nfor vI = 1 : n\r\n\tv = c(vI);\r\n\tu = c(mod(vI + 1, n) + n * (vI == n - 1));\r\n\t\r\n\tassert(findedge(g, v, u) \u003e 0);\r\nend\r\n\r\n%%\r\nn = 30;\r\np = 0.3;\r\nfor pI = 1 : 5\r\n\tn = n * 2;\r\n\tp = p / 2;\r\n\t\r\n\tx       = rand(n) \u003c p;\r\n\tx       = x + diag(ones(1, n - 1), -1);\r\n\tx(n, 1) = 1;\r\n\tx       = x + x.';\r\n\tr       = randperm(n);\r\n\tx(r, r) = x \u003e 0;\r\n\tg       = graph(x);\r\n\t\r\n\tc = findCycle(g);\r\n\tfor vI = 1 : n\r\n\t\tv = c(vI);\r\n\t\tu = c(mod(vI + 1, n) + n * (vI == n - 1));\r\n\t\t\r\n\t\tassert(findedge(g, v, u) \u003e 0);\r\n\tend\r\nend","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":692,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-03-24T02:36:00.000Z","updated_at":"2020-03-24T02:37:44.000Z","published_at":"2020-03-24T02:37:44.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou are given a graph g and asked to find a Hamiltonian cycle of g.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/matlab/ref/graph.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eMATLAB graph documentation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for details of the graph data structure.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA cycle of g is a sequence of vertices of g such that each adjacent pair of vertices in the sequence share an edge in g and the first and last vertices in the sequence share an edge in g. A Hamiltonian cycle of g is a cycle of g that visits each vertex of g exactly once.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, consider the adjacency matrix below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[A = [\\n   0     0     0     1     1     0     1     1\\n   0     1     0     1     0     0     1     0\\n   0     0     0     1     1     0     0     1\\n   1     1     1     1     0     1     0     1\\n   1     0     1     0     0     1     0     1\\n   0     0     0     1     1     0     0     1\\n   1     1     0     0     0     0     0     0\\n   1     0     1     1     1     1     0     0];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis corresponds to the graph with vertices labeled 1 through 8 and an edge between two vertices i and j if and only if A(i, j) == 1. This graph has cycles of vertices [3 4 8], [3 8 1 4], and [5 1 4 3 8], among others. Try the commands below to visualize this.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[g  = graph(A);\\ngh = plot(g);]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA Hamiltonian cycle for this graph g is [1 5 6 8 3 4 2 7].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor another fun challenge, see:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/45252-restricted-addition-v1\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eRestricted 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