On the existence of 5-Sun systems. (English) Zbl 1281.05106
Summary: A \(k\)-sun graph \(S(C_k)\) is a graph obtained from a \(k\)-cycle by adding a pendent edge to each vertex of the \(k\)-cycle. A \(k\)-sun system of order \(v\) is a decomposition of the complete graph \(K_v\) into \(k\)-sun graphs. In this paper, we find the necessary and sufficient condition for the existence of a 5-sun system of order \(v\). Moreover, we obtain cyclic 5-sun systems when \(n\equiv 1,5\) (mod 20) and 1-rotational 5-sun systems when \(n\equiv 0\) (mod 20).
MSC:
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
| 05C38 | Paths and cycles |
References:
| [1] | Anitha, R.; Lekshmi, R. S., \(N\)-sun decomposition of complete graphs and complete bipartite graphs, World Acad. Sci. Eng. Tech., 27, 262-266 (2007) |
| [2] | Anitha, R.; Lekshmi, R. S., \(N\)-sun decomposition of complete, complete bipartite and some Harary graphs, Int. J. Math. Sci., 2, 33-38 (2008) · Zbl 1186.05094 |
| [3] | Buratti, M., Rotational \(k\)-cycle systems of order \(v < 3 k\); another proof of the existence of odd cycle systems, J. Combin. Des., 11, 6, 433-441 (2003) · Zbl 1044.05045 |
| [4] | Buratti, M., Existence of 1-rotational \(k\)-cycle systems of the complete graph, Graphs Combin., 20, 1, 41-46 (2004) · Zbl 1064.05036 |
| [5] | Buratti, M.; Del Fra, A., Existence of cyclic \(k\)-cycle systems of the complete graphs, Discrete Math., 261, 1-3, 113-125 (2003) · Zbl 1013.05023 |
| [6] | Frucht, R., Graceful numbering of wheels and related graphs, Ann. New York Acad. Sci., 319, 219-229 (1979) · Zbl 0485.05059 |
| [7] | Fu, H.-L.; Wu, S.-L., Cyclically decomposing the complete graph into cycles, Discrete Math., 282, 1-3, 267-273 (2004) · Zbl 1042.05082 |
| [8] | Fu, Chin-Mei; Jhuang, Nan-Hua; Lin, Yuan-Lung; Sung, Hsiao-Ming, From Steiner triple systems to 3-sun systems, Taiwanese J. Math., 16, 2, 531-543 (2012) · Zbl 1242.05036 |
| [9] | Fu, Chin-Mei; Jhuang, Nan-Hua; Lin, Yuan-Lung; Sung, Hsiao-Ming, On the existence of \(k\)-sun systems, Discrete Math., 312, 1931-1939 (2012) · Zbl 1243.05197 |
| [10] | Grace, T., On sequential labellings of graphs, J. Graph Theory, 7, 195-201 (1983) · Zbl 0522.05063 |
| [11] | Harary, F., Graph Theory (1969), Addison-Wesley: Addison-Wesley New York · Zbl 0182.57702 |
| [12] | Liang, Z.; Guo, J., Decomposition of complete multigraphs into crown graphs, J. Appl. Math. Comput., 32, 507-517 (2010) · Zbl 1227.05193 |
| [13] | Liang, Z.; Guo, J.; Wang, J., On the crown graph decompositions containing odd cycle, Int. J. Combin. Graph Theory Appl., 2, 125-160 (2008) |
| [14] | Lindner, C. C.; Rodger, C. A., Design theory, (CRC Press Series on Discrete Mathematics and its Applications (1997)) · Zbl 0926.68090 |
| [15] | Mishima, M.; Fu, H.-L, Resolvable even-cycle systems with a 1-rotational automorphism, J. Combin. Des., 11, 394-407 (2003) · Zbl 1031.05074 |
| [16] | Wu, S.-L.; Fu, H.-L., Cyclic \(m\)-cycle systems with \(m \leq 32\) or \(m = 2 q\) with \(q\) a prime power, J. Combin. Des., 14, 66-81 (2006), 2006 · Zbl 1085.05040 |
| [17] | Wu, S.-L.; Lu, H.-C., Cyclically decomposing the complete graph into cycles with pendent edges, Ars Combin., 86 (2008) · Zbl 1224.05341 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
