Tearing a strip off the plane

In this article we examine some homomorphic properties of certain subgraphs of the unit-distance graph. We define G_r to be the subgraph of the unit-distance graph induced by the subset (-∞, ∞) × [0, r] of the plane. The bulk of the article is devoted to examining the graphs G_r, when r is the minimum width such that G_r contains an odd cycle of given length. We determine for each odd n the minimum width r_n such that $G_{r_{n}}$ contains an n-cycle C_n, and characterize the embeddings of C_n in $G_{r_{n}}$. We then show that $G_{r_{n}}$ is homomorphically equivalent to C_n when n ≡ 3 (mod 4), but $G_{r_{n}}$ is a core when n ≡ 1 (mod 4). We begin by showing that G_r is homomorphically compact for each r ≥ 0, as defined in [1]. We conclude with some other interesting results and open problems related to the graphs G_r. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 17–33, 1998