https://combinatorialpress.com/jcmcc-articles/volume-117/gregarious-y_5-tree-decompositions-of-tensor-product-of-complete-graphs/

Abstract

Yk-tree is defined as (v1,v2,…,vk−1;vk−2vk) by taking their vertices as (v1,v2,…,vk) and edges as {(v1v2,v2v3,…,vk−2vk−1)∪(vk−2vk)}. It is also represented as (Pk−1+e). One can obtain the necessary condition as mn(m−1)(n−1)≡0(mod2(k−1)), for k≥5 to establish a Yk-tree decomposition in Km×Kn. Here the tensor product is denoted by ×. In this anuscript, it is shown that a Y5-tree (gregarious Y5-tree) decomposition exists in Km×Kn, if and only if, mn(m−1)(n−1)≡0(mod8).