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 $n \geq 6$.