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Ternary Graph

A ternary graph is a graph with no vertex-induced subgraph that is a cycle graph of length divisible by 3 (Kim 2022, Zhang and Wu 2025, Bayer et al. 2026). Ternary graphs are also called trinity graphs (Zhang and Wu 2025). Every forest is ternary. A cycle graph C_n is ternary iff n is not divisible by 3, so C_4 and C_5 are ternary but C_3 and C_6 are not.

Kim (2022) characterizes these graphs by the topology of their independence complexes. A graph is ternary iff every nonempty vertex-induced subgraph has an independence complex that is contractible or homotopy equivalent to a sphere.

The term ternary graph is also used for a ternary diagram, which plots three proportions rather than representing vertices and edges.

A list of ternary graphs implemented in the Wolfram Language is available as GraphData["Ternary"].

See also

Chordless Cycle, Delannoy Polynomial, Independence Complex, Ternary Diagram

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References

Bayer, M.; Danner, R.; Holleben, T.; Kramer, M.; and Yang, Y. "Planar Ternary Graphs, Flag Spheres, and Delannoy Polynomials." Electron. J. Combin. 33, P3.66, 2026. https://doi.org/10.37236/14802.Kim, J. "The Homotopy Type of the Independence Complex of Graphs with No Induced Cycles of Length Divisible by 3." Europ. J. Combin. 104, 103534, 2022.Zhang, W. and Wu, H. "The Total Betti Number of the Independence Complex of Ternary Graphs." J. Eur. Math. Soc. 27, 269-278, 2025. https://doi.org/10.4171/JEMS/1378.

Referenced on Wolfram|Alpha

Ternary Graph

Cite this as:

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

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