Two graphs G1 and G2, each on n vertices, pack if there exists a bijection f from V (G1) onto V (G2) such that uv ∈ E(G1) only if f (u)f (v) ∉ E(G2). In 2014, Alon and Yuster proved that, for sufficiently large n, if |E(G1)| < n − δ(G2) and ∆(G2) ≤ √ n/200, then G1 and G2 pack. In this paper, we characterize the pairs of graphs for which the theorem of Alon and Yuster is sharp.We also prove the...

Background: The combination of doxorubicin (DOX) with paclitaxel (PTX) effectively treats breast cancer (BC). However,DOX-associated cardiotoxicity (CTX) is aggravated by the use PTX. Consensus lacking about which drug sequence involves most CTX. Objectives: To evaluate whether DOX followed PXT or reverse has greatest cardiotoxic potential in treatment BC. Methods: Prospective study women prima...

Let WQ = Proj(Q[g2' g3' X, Y, ZJI(homogeneous ideal generated by y 2Z + 4X3 g2XZ2 g3Z3». This is said to be the Weierstrass Family over the field Q. Then the first homology with compact supports of the Weierstrass Family is computed explicitly, i.e., it is generated by {C-kdX 1\ dYh .. , and {XC-kdX 1\ dYh .. 1 over the ring Q[g2' g3]' where C is a polynomial y 2 4X3 + g2 X + g3' When one tenso...

Let G1 and G2 be simple graphs and let n1 = |V (G1)|, m1 = |E(G1)|, n2 = |V (G2)| and m2 = |E(G2)|. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G1 G2 of G1 and G2. We show that: τ(G1 G2) ≥ 2(n1−1)(n2−1) n1n2 (τ(G1)n1) n2+1 2 (τ(G2)n2) n1+1 2 and τ(G1 G2) ≤ τ(G1)τ(G2) [ 2m1 n1 − 1 + 2m2 n2 − 1 ](n1−1)(n2−1) . We also characteri...