Taiwanese Journal of Mathematics

DECOMPOSITION OF COMPLETE GRAPHS INTO TRIANGLES AND CLAWS

Chin-Mei Fu, Yuan-Lung Lin, Shu-Wen Lo, and Yu-Fong Hsu

Full-text: Open access

Abstract

Let $K_n$ be a complete graph with $n$ vertices, $C_k$ denote a cycle of length $k$, and $S_k$ denote a star with $k$ edges. If $k=3$, then we call $C_3$ a triangle and $S_3$ a claw. In this paper, we show that for any nonnegative integers $p$ and $q$ and any positive integer $n$, there exists a decomposition of $K_n$ into $p$ copies of $C_3$ and $q$ copies of $S_3$ if and only if $3(p + q) = {n\choose 2}$, $q \ne 1, 2$ if $n$ is odd, $q = 1$ if $n=4$, and $q\ge \max \{3, \lceil\frac{n}{4}\rceil \}$ if $n$ is even and $n \geq 6$.

Article information

Source
Taiwanese J. Math., Volume 18, Number 5 (2014), 1563-1581.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706526

Digital Object Identifier
doi:10.11650/tjm.18.2014.3169

Mathematical Reviews number (MathSciNet)
MR3265077

Zentralblatt MATH identifier
1357.05099

Subjects
Primary: 05C51: Graph designs and isomomorphic decomposition [See also 05B30]

Keywords
graph decomposition complete graph cycle star

Citation

Fu, Chin-Mei; Lin, Yuan-Lung; Lo, Shu-Wen; Hsu, Yu-Fong. DECOMPOSITION OF COMPLETE GRAPHS INTO TRIANGLES AND CLAWS. Taiwanese J. Math. 18 (2014), no. 5, 1563--1581. doi:10.11650/tjm.18.2014.3169. https://projecteuclid.org/euclid.twjm/1499706526


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References

  • I. Anderson, Combinatorial Designs: Construction Methods, Ellis Horwood Limited, England, 1990.
  • J. C. Bermond, O. Favaron and M. Matheo, Hamiltonian decomposition of Cayley graphs of degree 4, J. Combin. Theory Ser. B, 46 (1989), 142-153.
  • J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan Press, London, 1976.
  • C. J. Colbourn, D. G. Hoffman and R. Rees, A new class of group divisible designs with block size three, J. Combin. Theory Ser. A, 59 (1992), 73-89.
  • T. P. Kirkman, On a problem in combinations, Cambridge and Dublin Math. Journal, 2 (1847), 191-204.
  • C. C. Lindner and C. A. Rodger, Design Theory, Boca Raton, Florida, CRC Press, 1997.
  • T. W. Shyu, Decomposition of complete graphs into cycles and stars, Graphs and Combinatorics, 2011, pp. 1-13, doi:10.1007/s00373-011-1105-3.
  • T. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand., 5 (1957), 57-68.
  • M. Tarsi, Decomposition of complete multigraph into stars, Discrete Math., 26 (1979), 273-278.
  • S. Yamamoto, H. Ikeda, S. Shige-ede, K. Ushio and N. Hamada, On claw decomposition of complete graphs and complete bipartite graphs, Hiroshima Math. J., 5 (1975), 33-42.