Windmill graph

Windmill graph
The Windmill graph Wd(5,4).
Vertices	(k-1)n+1
Edges	nk(k−1)/2
Radius	1
Diameter	2
Girth	3 if k > 2
Chromatic number	k
Chromatic index	n(k-1)
Notation	Wd(k,n)

In the mathematical field of graph theory, the windmill graph Wd(k,n) is an undirected graph constructed for k ≥ 2 and n ≥ 2 by joining n copies of the complete graph K_k at a shared vertex. That is, it is a 1-clique-sum of these complete graphs.^[1]

Properties

It has (k-1)n+1 vertices and nk(k−1)/2 edges,^[2] girth 3 (if k > 2), radius 1 and diameter 2. It has vertex connectivity 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is (k-1)-edge-connected. It is trivially perfect and a block graph.

Special cases

By construction, the windmill graph Wd(3,n) is the friendship graph F_n, the windmill graph Wd(2,n) is the star graph S_n and the windmill graph Wd(3,2) is the butterfly graph.

Labeling and colouring

The windmill graph has chromatic number k and chromatic index n(k-1). Its chromatic polynomial can be deduced form the chromatic polynomial of the complete graph and is equal to ${\displaystyle \prod _{i=0}^{k-1}(x-i)^{n}.}$

The windmill graph Wd(k,n) is proved not graceful if k > 5.^[3] In 1979, Bermond has conjectured that Wd(4,n) is graceful for all n ≥ 4.^[4] This is known to be true for n ≤ 22.^[5] Bermond, Kotzig, and Turgeon proved that Wd(k,n) is not graceful when k = 4 and n = 2 or n = 3, and when k = 5 and m = 2.^[6] The windmill Wd(3,n) is graceful if and only if n ≡ 0 (mod 4) or n ≡ 1 (mod 4).^[7]

Gallery

Small windmill graphs.

References

1. Gallian, J. A. "Dynamic Survey DS6: Graph Labeling." Electronic J. Combinatorics, DS6, 1-58, Jan. 3, 2007. .
2. Weisstein, Eric W. "Windmill Graph". MathWorld.
3. K. M. Koh, D. G. Rogers, H. K. Teo, and K. Y. Yap, Graceful graphs: some further results and problems, Congr. Numer., 29 (1980) 559-571.
4. J.C. Bermond, Graceful graphs, radio antennae and French windmills, Graph Theory and Combinatorics, Pitman, London (1979) 18-37.
5. J. Huang and S. Skiena, Gracefully labeling prisms, Ars Combin., 38 (1994) 225- 242.
6. J. C. Bermond, A. Kotzig, and J. Turgeon, On a combinatorial problem of antennas in radioastronomy, in Combinatorics, A. Hajnal and V. T. Sos, eds., Colloq. Math. Soc. János Bolyai, 18, 2 vols. North-Holland, Amsterdam (1978) 135-149.
7. J.C. Bermond, A. E. Brouwer, and A. Germa, "Systèmes de triplets et différences associées", Problèmes Combinatoires et Théorie des Graphes, Colloq. Intern. du CNRS, 260, Editions du Centre Nationale de la Recherche Scientifique, Paris (1978) 35-38.