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Graph factorization
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Unsolved problem in mathematics Conjecture: If <i>n</i> is odd and <i>k</i> ≥ <i>n</i>, then <i>G</i> is 1-factorable. If <i>n</i> is even and <i>k</i> ≥ <i>n</i> − 1 then <i>G</i> is 1-factorable. <a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">More unsolved problems in mathematics</a> <p>In <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, a factor of a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> <i>G</i> is a <a href="/facts/Spanning_subgraph/W0MqKkyE">spanning subgraph</a>, i.e., a subgraph that has the same vertex set as <i>G</i>. A <i>k</i>-factor of a graph is a spanning <i>k</i>-<a href="/facts/Regular_graph/j9dSoLE6">regular</a> subgraph, and a <i>k</i>-factorization partitions the edges of the graph into disjoint <i>k</i>-factors. A graph <i>G</i> is said to be <i>k</i>-factorable if it admits a <i>k</i>-factorization. In particular, a 1-factor is a <a href="/facts/Perfect_matching/LmtRYLgK">perfect matching</a>, and a 1-factorization of a <i>k</i>-regular graph is a <a href="/facts/Edge_coloring/AcdpUxCv">proper edge coloring</a> with <i>k</i> colors. A 2-factor is a collection of disjoint <a href="/facts/Cycle_(graph_theory)/Is44edrv">cycles</a> that spans all vertices of the graph. </p>