Decomposition of the complete bipartite graph with a 1-factor removed into paths and stars

Let P_k denote a path on k vertices, and let S_k denote a star with k edges. For graphs F, G, and H, a decomposition of F is a set of edge-disjoint subgraphs of F whose union is F. A (G,H)-decomposition of F is a decomposition of F into copies of G and H using at least one of each. In this paper, necessary and sufficient conditions for the existence of the (P_{k+1},S_k)-decomposition of the complete bipartite graph with a 1-factor removed are given

Decomposition of Complete Bipartite Multigraphs Into Paths and Cycles Having k Edges

We give necessary and sufficient conditions for the decomposition of complete bipartite multigraph Km,n(λ) into paths and cycles having k edges. In particular, we show that such decomposition exists in Km,n(λ), when λ ≡ 0 (mod 2), and k(p + q) = 2mn for k ≡ 0 (mod 2) and also when λ ≥ 3, λm ≡ λn ≡ 0(mod 2), k(p + q) =λ_mn, m, n ≥ k, (resp., m, n ≥ 3k/2) for k ≡ 0(mod 4) (respectively, for k ≡ 2(mod 4)). In fact, the necessary conditions given above are also sufficient when λ = 2

Decomposition of Complete Bipartite Multigraphs Into Paths and Cycles Having kk Edges

We give necessary and sufficient conditions for the decomposition of complete bipartite multigraph $K_{m,n} ( \lambda )$ into paths and cycles having $k$ edges. In particular, we show that such decomposition exists in $K_{m,n} ( \lambda )$, when $\lambda \equiv 0 (mod 2)$, m,n \geq k/2, m+n > k and $k(p + q) = 2mn$ for $k \equiv 0 (mod 2)$ and also when $\lambda \geq 3$, $\lambda m \equiv \lambda n \equiv 0(mod 2)$, $k(p + q) =\lambda m n$, $m, n \geq k$, (resp., $m, n \geq 3k//2$) for $k \equiv 0(mod 4)$ (respectively, for $k \equiv 2(mod 4)$). In fact, the necessary conditions given above are also sufficient when $\lambda = 2$