In the context of list coloring the vertices of a graph, Hall’s condition is a generalization of Hall’s Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list coloring. The graph G with list assignment L, abbreviated (G,L), satisfies Hall’s condition if for each subgraph H of G, the inequality |V (H)| 6σ∈C α(H(σ, L)) is satisfied, where C is the set of colors ...

Böttcher, Schacht and Taraz [6] gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós [15]. We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence...

We establish global Hölder regularity for eigenfunctions of the fractional g?Laplacian with Dirichlet boundary conditions where g=G? and G is a Young function satisfying so called ?2 condition. Our results apply to more general semilinear equations form (??g)su=f(u).

A graph G = (V, E) is called a split graph if there exists a partition V = I ∪K such that the subgraphs G[I ] and G[K] of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary condition for a split graph G with |I | < |K| to be hamiltonian. We will call a split graph G with |I | < |K| satisfying this condition a Burkard-Hammer graph. Furt...

Let G be a mixed graph. We discuss the relation between the second largest eigenvalue λ2(G) and the second largest degree d2(G), and present a sufficient condition for λ2(G) ≥ d2(G).

Abstract G-Brownian motion has potential applications in uncertainty problems and risk measures, which attracted the attention of many scholars. This study investigates almost sure exponential stability nonlinear stochastic delay hybrid systems driven by motion. Due to non-linearity G-expectation distribution motion, it is difficult this issue. Firstly, existence global unique solution derived ...

Vizing's theorem states that the chromatic index 0 (G) of a graph G is either the maximum degree (G) or (G) + 1. A graph G is called overfull if jE(G)j > (G)bjV (G)j=2c. A suucient condition for 0 (G) = (G) + 1 is that G contains an overfull subgraph H with (H) = (G). Plantholt proved that this condition is necessary for graphs with a universal vertex. In this paper, we conjecture that, for ind...

where the terminal condition ξ and the coefficient g = g(t, y, z) are given. W is a d–dimensional Brownian motion. The solution (yt, zt)t∈[0,T ] is a pair of square integrable processes. A foundational and interesting problem is: what is the relationship between the uniqueness of solution and continuous dependence with respect to g or ξ? In the standard situation where g satisfies linear growth...

We provide a sufficient condition for the cyclicity of an ordered tensor product L = Va1(ωb1) ⊗ Va2(ωb2) ⊗ . . . ⊗ Vak(ωbk) of fundamental representations of the Yangian Y (g). When g is a classical simple Lie algebra, we make the cyclicity condition concrete, which leads to an irreducibility criterion for the ordered tensor product L. In the case when g = sll+1, a sufficient and necessary cond...

This paper studies the subexponential asymptotics of the stationary distribution of an M/G/1-type Markov chain. We provide a sufficient condition for the subexponentiality of the stationary distribution. The sufficient condition requires only the subexponential integrated tail of level increments. On the other hand, the previous studies assume the subexponentiality of level increments themselve...