## Taiwanese Journal of Mathematics

### DECOMPOSITION OF COMPLETE GRAPHS INTO TRIANGLES AND CLAWS

#### 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

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

#### 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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