2025/07/21 by Liu, Jinhao, Lü, Huazhong
#05C38 #68R10 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.15463
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≤ k≤ 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.