Cycles Embedding in Exchanged Crossed Cube

Abstract

The (s+t+1)-dimensional exchanged crossed cube, denoted by ECQ(s, t), proposed by Li et al., combines the advantages of the hypercube and the crossed cube. It has been proven that ECQ(s, t) has better properties than the fundamental hypercube in the aspects of the fewer edges, lower cost factor and smaller diameter. This paper studies the embedding of cycles in ECQ(s, t). It is proved that ECQ(s, t) contains an l-cycle of every length l from 4 to 2s+t+1 except that ECQ(2, 3) and ECQ(3, 3) do not contain a cycle of length 9 where [Formula: see text] and [Formula: see text]. This result reveals the fact that ECQ(s, t) nearly remains the cycle embedding capability, while it only has about half edges of crossed cube.