Covering graph

In the mathematical discipline of graph theory, a graph $C$ is a covering graph of another graph $G$ if there is a covering map from the vertex set of $C$ to the vertex set of $G$ . A covering map $f$ is a surjection and a local isomorphism: the neighbourhood of a vertex $v$ in $C$ is mapped bijectively onto the neighbourhood of ⁠ $f(v)$ ⁠ in $G$ .

The term lift is often used as a synonym for a covering graph of a connected graph.

Though it may be misleading, there is no (obvious) relationship between covering graph and vertex cover or edge cover.

The combinatorial formulation of covering graphs is immediately generalized to the case of multigraphs. A covering graph is a special case of a covering complex.^[1] Both covering complexes and multigraphs with a 1-dimensional cell complex, are nothing but examples of covering spaces of topological spaces, so the terminology in the theory of covering spaces is available; say covering transformation group, universal covering, abelian covering, and maximal abelian covering.^[2]

^ Rotman, Joseph (December 1973). "Covering complexes with applications to algebra". Rocky Mountain Journal of Mathematics. 3 (4): 641–674. doi:10.1216/RMJ-1973-3-4-641.
^ Sunada, Toshikazu (2012). "6 Topological Crystals". Topological Crystallography: With a View Towards Discrete Geometric Analysis. Springer. pp. 73–90. ISBN 978-4-431-54177-6.

[1] Rotman, Joseph (December 1973). "Covering complexes with applications to algebra". Rocky Mountain Journal of Mathematics. 3 (4): 641–674. doi:10.1216/RMJ-1973-3-4-641.

[sunada-2] Sunada, Toshikazu (2012). "6 Topological Crystals". Topological Crystallography: With a View Towards Discrete Geometric Analysis. Springer. pp. 73–90. ISBN 978-4-431-54177-6.

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