Odd cycle

An odd cycle is a closed path containing an odd number of edges, with no repeated vertices except its starting and ending vertex.

Graph theory: The mathematical study of graphs, structures made of vertices joined by edges. Odd cycles are defined and studied as structures inside graphs.
Bipartite graph: A graph whose vertices can be divided into two sets so every edge joins different sets. A graph is bipartite exactly when it contains no odd cycle.
Matching (graph theory): A set of edges in a graph no two of which share a vertex. Odd cycles complicate the search for maximum matchings in general graphs.
Even cycle: A cycle containing an even number of edges. Even cycles permit consistent alternation between two vertex classes.
Graph (discrete mathematics): A structure consisting of vertices and edges that connect pairs of vertices. An odd cycle is a substructure formed from a graph’s vertices and edges.
Graph coloring: The assignment of colors to graph elements subject to specified adjacency constraints. An odd cycle requires three colors for a proper vertex coloring.
Blossom algorithm: An algorithm for finding maximum matchings in general graphs by contracting odd alternating cycles. Its blossom structures are odd cycles that obstruct simpler augmenting-path searches.
Chord (graph theory): An edge joining two nonconsecutive vertices of a cycle. A chord splits an odd cycle into smaller cycles, changing its induced structure.
Walk (graph theory): A sequence of vertices in a graph where consecutive vertices are joined by edges. A cycle is a closed walk with additional restrictions on repeated vertices.
Two-coloring: An assignment of two colors to vertices so adjacent vertices receive different colors. Alternating colors around an odd cycle forces its endpoints to conflict.

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