thumb|upright=1.35|An '''edge cycle cover''' for each graph. The covering of the middle graph is ''edge-disjoint'', while the one of the right graph is ''vertex-disjoint''.
In graph theory, a branch of mathematics, an '''edge cycle cover''' (sometimes called simply '''cycle cover'''<ref>Cun-Quan Zhang, Integer flows and cycle covers of graphs, Marcel Dekker,1997.</ref>) of a graph is a family of cycles which are subgraphs of ''G'' and contain all edges of ''G''.
If the cycles of the cover have no vertices in common, the cover is called '''vertex-disjoint''' or sometimes simply '''disjoint cycle cover'''. In this case, the set of the cycles constitutes a spanning subgraph of ''G''.
If the cycles of the cover have no edges in common, the cover is called '''edge-disjoint''' or simply '''disjoint cycle cover'''.
==Properties and applications==
===Minimum-Weight Cycle Cover=== For a weighted graph, the Minimum-Weight Cycle Cover Problem (MWCCP) is the problem to find a cycle cover with minimal sum of weights of edges in all cycles of the cover.
For bridgeless planar graphs, the MWCCP can be solved in polynomial time.<ref>"Handbook in Graph Theory" (2004) {{isbn|1-58488-090-2}}, [https://books.google.com/books?id=mKkIGIea_BkC&dq=%22minimum+weight+cycle+cover%22&pg=PA225 p. 225]</ref>
==Cycle ''k''-cover== A '''cycle ''k''-cover''' of a graph is a family of cycles which cover every edge of ''G'' exactly ''k'' times. It has been proven that every bridgeless graph has cycle ''k''-cover for any even integer ''k''≥4. For ''k''=2, it is the well-known cycle double cover conjecture is an open problem in graph theory. The cycle double cover conjecture states that in every bridgeless graph, there exists a set of cycles that together cover every edge of the graph twice.<ref>{{Cite web |url=http://www.cems.uvm.edu/%7Earchdeac/problems/cyclecov.htm |title="The Cycle Double Cover Conjecture" |access-date=2008-12-21 |archive-date=2011-07-20 |archive-url=https://web.archive.org/web/20110720105312/http://www.cems.uvm.edu/%7Earchdeac/problems/cyclecov.htm |url-status=dead }}</ref>
==See also== *Alspach's conjecture *Vertex cycle cover
==References== {{reflist}}
Category:Graph theory objects Category:Combinatorial optimization
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