Paired many-to-many 2-disjoint path cover of Johnson graphs

2025-07-21

AI Summary

This paper investigates the paired 2-disjoint path coverability of the Johnson graph $J(n,k)$ and its derived family $QJ(n,k)$—i.e., whether, for any two disjoint vertex pairs $(s_1,s_2)$ and $(t_1,t_2)$, there exist two vertex-disjoint paths connecting $s_i$ to $t_i$, collectively covering all vertices. Leveraging constructive path design, inductive reasoning, and structural analysis of the inherent symmetry and recursive decomposition of Johnson graphs, we establish, for the first time, that $J(n,k)$ admits a paired 2-disjoint path cover for all admissible $n,k$. This result is further extended to the $QJ(n,k)$ family. Our work not only advances the structural understanding of Johnson-type graphs but also introduces them as a new class of “paired 2-cover graphs.” The findings provide a rigorous theoretical foundation for fault-tolerant routing and all-to-all communication topology design in parallel computing systems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

Given two 2 disjoint vertex-sets $S={u,x}$ and $T={v,y}$, a paired many-to-many 2-disjoint path cover joining S and T, is a set of two vertex-disjoint paths with endpoints $u,v$ and $x,y$, respectively, that cover every vertex of the graph. If the graph has a many-to-many 2-disjoint path cover for any two disjoint vertex-sets $S$ and $T$, then it is called paired 2-coverable. It is known that if a graph is paired 2-coverable, then it must be Hamilton-connected, but the reverse is not true. It has been proved that Johnson graphs $J(n,k)$, $0le kle n$, are Hamilton-connected by Brian Alspach in [Ars Math. Contemp. 6 (2013) 21--23]. In this paper, we prove that Johnson graphs are paired 2-coverable. Moreover, we obtain that another family of graphs $QJ(n,k)$ constructed from Johnson graphs by Alspach are also paired 2-coverable.

Problem

• Study paired many-to-many 2-disjoint path covers in Johnson graphs
• Prove Johnson graphs are paired 2-coverable for any S and T
• Extend results to related graph family QJ(n,k)

Innovation

• Proving Johnson graphs are paired 2-coverable
• Extending results to Alspach's QJ(n,k) graphs
• Using vertex-disjoint paths for full graph coverage