Closed trail decompositions on grid graphs

Let G = G(n, m) be a rectangular solid grid graph and 𝒜(G) be a minimum length Eulerian augmentation of G. Let l₀, . . ., l_t ∈ ℕ such that ∑^t_i=0 l_i = |E(𝒜(G)|, where 2(n + m) ≤ l_i = 2k_i. In this paper, we exhibit a constructive procedure providing an edge-disjoint decomposition of 𝒜 (G) into closed trails T₀, . . ., T_t such that |E(T_i)| = l_i.