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...

Koh and Tan gave a sufficient condition for a 3-partite tournament to have at least one 3-king in [K.M. Koh, B.P. Tan, Kings in multipartite tournaments, Discrete Math. 147 (1995) 171–183, Theorem 2]. In Theorem 1 of this paper, we extend this result to n-partite tournaments, where n 3. In [K.M. Koh, B.P. Tan, Number of 4-kings in bipartite tournaments with no 3-kings, Discrete Math. 154 (1996)...

We generalize two-graphs (\unitogs") to \togs", structures consisting of sets of polygons or cycles in a graph that satisfy speciied relations. Togs are equivalent to switching classes of signings of a particular base graph. We develop a mechanism for generating and proving examples and apply it to unitogs (based on complete graphs), bipartite, tripartite, and multi-partite togs (based on compl...

An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs H and G , the anti-Ramsey number a r ( ) in maximum colors coloring so that there does not exist copy . Li et al. determined k -matchings complete bipartite graphs. Jin Zang showed uniqueness extremal coloring. In this paper, as generalization these results, we determine K n 1 … M -partite -un...

If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D) = max{d+(x), d−(x)}−min{d+(y), d−(y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is almost regular. A c-partite tournament is an orientation of a complete c-...

As usual λ(G) denotes the edge-connectivity of graph G. It was shown in [2] that every G contains a spanning (λ(G)+1)-partite subgraph H such λ(H)=λ(G) and one can find polynomial time. We determine complexity deciding, for given positive integers r,k whether r-colourable which is k-edge-connected. show problem polynomially solvable when r>k NP-complete otherwise. In fact, combined with result ...

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.

A classical theorem of Simonovits from the 1980s asserts that every graph $G$ satisfying ${e(G) \gg v(G)^{1+1/k}}$ must contain $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$ copies $C_{2k}$. Recently, Morris and Saxton established a balanced version Simonovits' theorem, showing such has $C_{2k}$, which are `uniformly distributed' over edges $G$. Moreover, they used this result to obtain sharp b...

It is known that every hypercube Qn is a bipartite graph. Assume that n 2 and F is a subset of edges with |F | n− 2. We prove that there exists a hamiltonian path in Qn − F between any two vertices of different partite sets. Moreover, there exists a path of length 2n − 2 between any two vertices of the same partite set. Assume that n 3 and F is a subset of edges with |F | n− 3. We prove that th...