Edge-Bipancyclicity of Bubble-Sort Star Graphs

The topology of interconnection networks determines the performance of the networks. Linear arrays and rings are two of the most fundamental structures of the interconnection network topologies owing to their simple structures and low degree. Thus how to embed cycles and paths into interconnection networks is a crucial factor for the networks. The interconnection network considered in this paper is the bubble-sort star graph. The
$n$
-dimensional bubble-sort star graph
$BS_{n}$
is a bipartite and
$(2n- 3)$
-regular graph of order
$n!$
. A bipartite graph
$G$
of order
$|V(G)|$
is edge-bipancyclic if each edge of
$G$
lies on a cycle of all even length
$l$
with
$4\leq l\leq |V(G)|$
. In this paper, we show that the
$n$
-dimensional bubble-sort star graph
$BS_{n}$
is edge-bipancyclic for
$n\ge 3$
and for each even length
$l$
with
$4\leq l\leq n!$
, every edge of
$BS_{n}$
lies on at least four different cycles of length
$l$
. Moreover, we also have that
$BS_{n}$
is vertex-bipancyclic and bipancyclic for
$n\ge 3$
.