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

TL;DR

This paper proves that Johnson graphs and certain related graphs are paired 2-coverable, meaning they can be covered by two disjoint paths connecting any two disjoint vertex pairs, extending known Hamilton-connectivity results.

Contribution

It establishes that Johnson graphs and related graphs are paired 2-coverable, a property stronger than Hamilton-connectivity, with implications for graph covering and path decomposition.

Findings

Johnson graphs are paired 2-coverable.

Related graphs QJ(n,k) are also paired 2-coverable.

Extends Hamilton-connectivity results to path cover properties.

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 $J(n,k)$, $0\le k\le n$, 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.