Path covers of bubble-sort star graphs

Abstract

The distributed computing or parallel computing system uses an interconnection network as a topology structure to connect a large number of processors. The disjoint paths of interconnection networks are related to parallel computing and the fault tolerance. l-path cover of graph $G=(V(G), E(G))$ consists of (internally) disjoint paths $P_k$s ($1 \le k \le l$), where $\cup _{k=1}^lV(P_k)=V(G)$. The bubble-sort star graph is bipartite and has favorable reliability and fault tolerance which are critical for multiprocessor systems. We focus on the one-to-one 1-path cover, one-to-one $(2n-3)$-path cover, and many-to-many 2-path cover of the bubble-sort star graph $BS_n$. More specifically, let $V(BS_n)=V_e \cup V_o$ with $V_e \cap V_o=\emptyset$, for $\{u, x\} \subset V_e$ and $\{v, y\} \subset V_o$, we prove that (1) $BS_n$ contains a 1-path cover, i.e., Hamiltonian path $P_{uv}$, (2) $BS_n$ contains one-to-one $(2n-3)$-path cover $P_k$s ($1 \le k \le 2n-3$) between u and v, and (3) $BS_n$ contains many-to-many 2-path cover $P_{uv}$ and $P_{xy}$, where $n \ge 3$. Since $BS_n$ is $(2n-3)$-regular graph, the one-to-one $(2n-3)$-path cover is the maximal one-to-one path cover.