In this paper, we prove some Bernstein type results for n-dimensional minimal Lagrangian graphs in quaternion Euclidean space Hn ∼= R. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in Cn.

In this note we prove the following conjecture of Nowakowski and Rall: For arbitrary graphs G and H the upper domination number of the Cartesian product G H is at least the product of their upper domination numbers, in symbols: Γ(G H) ≥ Γ(G)Γ(H). A conjecture posed by Vizing [7] in 1968 claims that Vizing’s conjecture: For any graphs G and H, γ(G H) ≥ γ(G)γ(H), where γ, as usual, denotes the do...

In a digraph D = (X,U) , not necessarily finite, an arc (x, y) ∈ U is reachable from a vertex u if there exists a directed walk W that originates from u and contains (x, y) . A subset S ⊆ X is an arc-reaching set of D if for every arc (x, y) there exists a diwalk W originating at a vertex u ∈ S and containing (x, y) . A minimal arc-reaching set is an arc-basis. S is a point-reaching set if for ...

We consider the fault hamiltonian properties of m×n meshes with two wraparound edges in the first row and the last row, denoted by M2(m, n), m ≥ 2, n ≥ 3. M2(m, n) is a spanning subgraph of Pm × Cn which has interesting fault hamiltonian properties. We show that M2(m, n) with odd n is hamiltonian-connected and 1-fault hamiltonian. For even n, M2(m, n), which is bipartite, with a single faulty e...

A colouring of the edges of an n×n grid is said to be reconstructible if the colouring is uniquely determined by the multiset of its n tiles, where the tile corresponding to a vertex of the grid specifies the colours of the edges incident to that vertex in some fixed order. In 2015, Mossel and Ross asked the following question: if the edges of an n× n grid are coloured independently and uniform...

We numerically investigate the epidemic spread phenomena and efficient immunization strategies on complex networks embedded in geometry. It is assumed that there exists an unavoidable time delay (we call it the detection time) between the actual infection and the beginning of immunization, and we implement two different immunization strategies: one is based on topological connection neighbors (...

i∈F ci ≥ k for all F ∈ F(∆). If c is a (0, 1)-vector, then c may be identified with the subset C = {i ∈ [n] : ci 6= 0} of [n]. It is clear that c is a 1-cover if and only if C is a vertex cover of ∆ in the classical sense, that is, C ∩ F 6= ∅ for all F ∈ F(∆). Let S = K[x1, . . . , xn] be a polynomial ring in n variables over a field K. Let Ak(∆) denote the K-vector space generated by all monom...