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# Unicyclic Graph

A unicyclic graph is a connected graph containing exactly one cycle (Harary 1994, p. 41). A connected unicyclic graph is therefore a pseudotree that is not a tree.

Truszczyński (1984) conjectured that all unicyclic graphs except the cycle graph with or 2 (mod 4) are graceful (Gallian 2018).

The numbers of unicyclic graphs on , 2, ... vertices are 0, 0, 1, 3, 9, 25, 68, 185, ... (OEIS A236570), the first few of which are illustrated above.

The corresponding numbers of connected unicyclic graphs are 0, 0, 1, 2, 5, 13, 33, 89, ... (OEIS A001429), the first few of which are illustrated above.

Examples of unicyclic classes of graphs include -caveman graphs, cycle graphs , pan graphs, sunlet graphs , and tadpole graphs.

Acyclic Graph, Cyclic Graph, Graph Cycle, Pseudotree, Tree

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## References

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.Sloane, N. J. A. Sequences A001429/M1438 and A236570 in "The On-Line Encyclopedia of Integer Sequences."Truszczyński, M. "Graceful Unicyclic Graphs." Demonstatio Math. 17, 377-387, 1984.

Unicyclic Graph

## Cite this as:

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