Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of H-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph H. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical H-...

For an integer t and a fixed graph H, we consider the problem of finding a maximum tmatching not containing H as a subgraph, which we call the H-free t-matching problem. This problem is a generalization of the problem of finding a maximum 2-matching with no short cycles, which has been well-studied as a natural relaxation of the Hamiltonian circuit problem. When H is a complete graph Kt+1 or a ...

We examine the influence of heterogeneous curing rates for a SIS model, used for malware spreading on the Internet, information dissemination in unreliable networks, and propagation of failures in networks. The topology structures considered are the regular graph which represents the homogenous network structures and the complete bi-partite graph which represents the hierarchical network struct...

We give a new separability criterion, a necessary condition for separability of N partite quantum states. The criterion is based on the Bloch representation of a N -partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N -partite quantum states in H = H1 ⊗H2 ⊗ · · ·⊗HN . The criterion can test whether a N -parti...

The problem of finding good lower bounds on the size of the largest bipartite subgraph of a given graph has received a fair amount of attention. In particular, improving a result of Erdős ([10]; see also [11] for related problems), Edwards [9] proved the essentially best possible assertion that every graph with n vertices and m edges has a bipartite subgraph with at least m/2 + (n − 1)/4 edges....

An equitable k-coloring of a graph G is proper such that the sizes any two color classes differ by at most one. (q,r)-tree-coloring an q-coloring subgraph induced each class forest maximum degree r. Let strong vertex r-arboricity G, denoted var≡(G), be minimum p has for every q≥p. The values va1≡(Kn,n) were investigated Tao and Lin Wu, Zhang, Li where exact found in some special cases. In this ...

We investigate coined quantum-walk search and state-transfer algorithms, focusing on the complete $M$-partite graph with $N$ vertices in each partition. First, it is shown that by adding a loop to vertex, algorithm finds marked vertex unit probability limit of large graph. Next, we employ evolution operator two perform state transfer between sender receiver. show when receiver are different par...

Graph embedding is a powerful method in parallel computing that maps guest network G into host H . The performance of an can be evaluated by certain parameters, such as the dilation, edge congestion, and wirelength. In this manuscript, we obtain wirelength (exact minimum) complete multi-partite graphs Cartesian product paths and/or cycles, which include n -cube, -dimensional mesh (grid), cylind...

For a fixed r, let fr(n) denote the minimum number of complete r-partite rgraphs needed to partition the complete r-graph on n vertices. The Graham-Pollak theorem asserts that f2(n) = n − 1. An easy construction shows that fr(n) 6 (1 + o(1)) ( n br/2c ) , and we write cr for the least number such that fr(n) 6 cr(1 + o(1)) ( n br/2c ) . It was known that cr < 1 for each even r > 4, but this was ...