Dimensions of Mycielskians of cycles

Abstract

The Mycielskian is a standard construction studied in many an introductory graph theory course. It is natural to consider Mycielskians of cycles, some of the simplest of all graphs. This paper deals with the so-called ``dimension'' of such graphs. The dimension of a graph $G$ is the smallest positive integer $n$ such that there exists a one-to-one correspondence between the vertices of $G$ and some collection of points in $n$-dimensional Euclidean space such that if two vertices in $G$ are adjacent, then the distance between the corresponding points is $1$. In previous works, it had been proven that the dimension of the Mycielskian of a $k$-cycle is 3 when $k$ is $3$, $4$, or $5$, and 2 when $k=10$. In this paper, we answer the question completely. Namely, we show that the dimension is $3$ when $k\neq 10$, and $2$ when $k=10$.