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  "body": "\n## Statement\n\nLet $G$ be a finite [[graph|00C0BR7D7H]], possibly with multiple edges, that has no [[bridge|0BANA3XYPF]].\nThe conjecture: there is a finite list of [[cycles|0Z05VCBQ7Y]] of $G$ such that each edge of $G$ lies in exactly two cycles of the list.\nThe list may contain a cycle more than once.\nSuch a list is called a cycle double cover of $G$.\n\n## Remarks\n\nThe conjecture is open.\nIt was posed by Szekeres and, independently, by Seymour.\nThe hypothesis is necessary: a bridge lies on no cycle, so a graph with a bridge has no cycle double cover.\nFor a bridgeless planar graph, each face boundary of a plane embedding splits into edge-disjoint cycles, and each edge lies on exactly two faces.\nSo these cycles form a cycle double cover.\nThe general case reduces to cubic bridgeless graphs, and further to snarks, the cubic bridgeless graphs with no proper edge colouring with $3$ colours.\nIf a cubic graph has a proper edge colouring with $3$ colours, the unions of two colour classes give three families of disjoint cycles, which together cover each edge twice.\n\n## Explanations\n\n### L1 human\n\nTake a network with no single link whose removal disconnects it.\nThe conjecture says that one can choose a collection of closed loops in the network that passes over every link exactly twice.\nFor a map drawn in the plane, the borders of the countries do this.\nFor general networks, nobody knows a proof, and nobody has found a counterexample.\n",
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      "posed_by": "George Szekeres and Paul Seymour"
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    "kind": "conjecture",
    "msc": [
      "05C38",
      "05C70"
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    "name": "Cycle double cover conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cycle_double_cover"
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    "tag": "G2RYQMGG1R",
    "uses": [
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  "link": "https://math2.tamnd.com/node/G2RYQMGG1R/",
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  "markdown": "https://math2.tamnd.com/node/G2RYQMGG1R.md",
  "msc": [
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    "05C70"
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  "name": "Cycle double cover conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/G2R/G2RYQMGG1R.md",
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    "tag": "G2RYQMGG1R",
    "kind": "conjecture",
    "level": "W0",
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