Sparse Graphs which Decompose into Closed Trails of Arbitrary Lengths

Abstract

A simple, connected even graph G with vertex set V(G) and edge set E(G) is said to be ADCT (Arbitrarily Decomposable into Closed Trails) if for any collection of positive integers x ₁, x ₂,...,x _m with $\sum_{i=1}^m x_i = |E(G)|$ and x _i ≥ 3 for 1 ≤ i ≤ m, there exists a decomposition of G into closed trails (circuits) of lengths x ₁, x ₂,...,x _m . In this note we construct an 8-regular ADCT graph on 6n vertices, for each each n ≥ 3. On the other hand, we also show that there are only finitely many 4-regular graphs which are ADCT.