Hamilton Paths in Certain Arithmetic Graphs

For each integer m 1, consider the graph Gm whose vertex set is the set N = {0, 1, 2, . . . } of natural numbers and whose edges are the pairs xy with y = x + m or y = x - m or y = mx or y = x/m. Our aim in this note is to show that, for each m, the graph Gm contains a Hamilton path. This answers a question of Lichiardopol. For each integer m 1, consider the graph Gm whose vertex set is the set N = {0, 1, 2, . . . } of natural numbers and whose edges are the pairs xy with y = x + m or y = x - m or y = mx or y = x/m. We show that, for each m, the graph Gm contains a Hamilton path. Here, by `Hamilton path' we mean a `oneway infinite Hamilton path', i.e. a sequence x0, x1, x2, . . . of vertices of Gm such that each vertex appears precisely once and, for all i, the vertices xi and xi+1 are adjacent. We shall use this to answer a question of Lichiardopol [1] about two-way infinite Hamilton paths in graphs defined similarly but with vertex set the set Z of integers. The case m =...

• Type: Journal
• Year: 2005
• Where: ARSCOM
• Authors: Paul A. Russell
• Keywords: ARSCOM 2005, Graph G2, Graph Gm, Hamilton Path