Graph Genus

The genus of a graph , denoted or , is the minimum number of handles that must be added to the plane to embed the graph without any crossings. A graph with genus 0 is embeddable in the plane and is said to be a planar graph. The names of graph classes having particular values for their genera are summarized in the following table (cf. West 2000, p. 266).

Every graph has a genus (White 2001, p. 53).

The smallest double-toroidal graph have 8 vertices, and there are precisely 15 double-toroidal graphs on 8 nodes.

There are no pretzel graphs on eight or fewer vertices.

Let be a surface of genus . Then if for , then has in embedding in but not in for . Furthermore, embeds in for all , as can be seen by adding handles to an embedding of in (White 2001, p. 52).

The genus of a graph satisfies

where is the edge count of (White 2001, p. 53).

The genus of a disconnected graph is the sum of the genera of its connected components (Battle et al. 1962, White 2001, p. 55), and the genus of a connected graph is the sum of the genera of its blocks (Battle et al. 1962; White 2001, p. 55; Stahl 1978).

It follows from results of Battle et al. (1962) that the disjoint union of copies of (or disjoint copies of is a forbidden minor for graphs of genus . Similarly, copies of or such that some share a vertex and which have blocks that are or are forbidden minors for graphs of genus .

Duke and Haggard (1972; Harary et al. 1973) gave a criterion for the genus of all graphs on 8 and fewer vertices. Define the double-toroidal graphs

where denotes minus the edges of . Then for any subgraph of :

1. if does not contain a Kuratowski graph (i.e., or ),

2. if contains a Kuratowski graph (i.e., is nonplanar) but does not contain any for ,

3. if contains any for .

The complete graph has genus

for , where is the ceiling function (Ringel and Youngs 1968; White 1973; Harary 1994, p. 118; Asir and Chelvam 2014). The values for , 2, ... are 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, ... (OEIS A000933).

The complete bipartite graph has genus

(Ringel 1965; Beineke and Harary 1965; White 1973; Harary 1994, p. 119; Asir and Chelvam 2014).

White conjectured that complete tripartite graph with has genus

(Stahl and White 1976, Ellingham et al. 2005, Ellingham et al. 2006). While this has not been proven in general, it has been established in a number of special cases (White 1969b, Stahl and White 1976, Craft 1991, Craft 1998, Ellingham et al. 2005, Ellingham et al. 2006), including the case

(White 1973, Asir and Chelvam 2014).

The complete 4-partite graph has genus

(White 1973, Asir and Chelvam 2014).

The hypercube has genus

(Ringel 1955; Beineke and Harary 1965; Harary et al. 1988; Harary 1994, p. 119). The values for , 2, ... are 0, 0, 0, 1, 5, 17, 49, 129, ... (OEIS A000337).

See also

Circuit Rank, Double-Toroidal Graph, Graph Crossing Number, Planar Graph, Pretzel Graph, Rectilinear Crossing Number, Toroidal Graph

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References

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Graph Genus

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Weisstein, Eric W. "Graph Genus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GraphGenus.html

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