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Kayak Paddle Graph

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KayakPaddleGraphs

A kayak paddle graph KP(k,m,l) is the graph obtained by joining cycle graphs C_k and C_m by a path of length l (Gallian 2018). A number of kayak paddle graphs are illustrated above.

Kayak paddle graphs are implemented in the Wolfram Language for small values of k, m, and l as GraphData[{"KayakPaddle", {k, m, l}}].

KP(3,3,1) is isomorphic to the 3-barbell graph.

Kayak paddle graphs are planar, cactus, unit-distance and matchstick graphs. They are also bridged and traceable and have arboricity of 2.

Litersky (2011) proved that kayak paddle graphs are graceful when:

1. k=0 (mod 4), m=0,3 (mod 4),

2. k=m=2 (mod 4) for k>=3,

3. k=1 (mod 4), m=3 (mod 4)

(Litersky 2011, Gallian 2018).

See also

Barbell Graph, Cycle Graph, Lollipop Graph, Pan Graph, Tadpole Graph

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References

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin., Dynamic Survey DS6, Oct. 30, 2025. https://doi.org/10.37236/27.House of Graphs. Kayak Paddle Graphs. 3-barbell graph, (3,3,2)-kayak paddle graph, (3,5,2)-kayak paddle graph, (3,5,4)-kayak paddle graph, (4,4,1)-kayak paddle graph, 4-Cycles with Vertex Between, and connected pentagons.Litersky, A. "Graceful Kayak Paddles." Master's thesis. Duluth, MN: University of Minnesota Duluth, 2011.

Referenced on Wolfram|Alpha

Kayak Paddle Graph

Cite this as:

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

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