Tearing a strip off the plane

Abstract

In this article we examine some homomorphic properties of certain subgraphs of the unit-distance graph. We define Gr 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 Gr, when r is the minimum width such that Gr contains an odd cycle of given length. We determine for each odd n the minimum width rn such that contains an n-cycle Cn, and characterize the embeddings of Cn in $G_{r_{n}}$. We then show that is homomorphically equivalent to Cn when n ≡ 3 (mod 4), but is a core when n ≡ 1 (mod 4). We begin by showing that Gr 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 Gr. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 17–33, 1998