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 suu-cient 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...

Vizing’s conjecture on the Cartesian product G H of two graphs, γ(G H) ≥ γ(G)γ(H), will hold for all graphs G if the domination and (closed neighborhood) packing number of G are equal. By using the theory of integer programming, we state known sufficient conditions for when the packing number and the domination number of a graph are equal. We also give a sufficient condition using the theory of...

and which satisfies the classical sign condition g(x,s,ξ)s ≥ 0. The right-hand side f is assumed to belong to W−1EM(Ω). It is well known that Gossez [12] solved (1.1) in the case where g depends only on x and u. If g depends also on ∇u, existence theorems have recently been proved by Benkirane and Elmahi in [3, 4] by making some restrictions. In [3], g is supposed to satisfy a “nonnatural” grow...

In this paper, we propose a sufficient condition for the existence of solutions to general variational inequality problems (GV I(K,F , g)). The condition is also necessary when F is a g-P ∗ function. We also investigate the boundedness of the solution set of (GV I(K,F, g)). Furthermore, we show that when F is norm-coercive, the general complementarity problems (GCP (K,F, g)) has a nonempty comp...

a {em roman dominating function} on a graph $g$ is a function$f:v(g)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}a {em restrained roman dominating}function} $f$ is a {color{blue} roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} the wei...

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condit...

JBSTRACT In the real algebra R(G) over a certain finite group, the operator exp is defined. A condition is stated on exp J which is necessary and sufficient for J in R(G) to be nonnegative (where J is viewed as a function G + R). Physically, this amounts to a condition on the correlations of a generalized system of Ising spins which is necessary and sufficient for the ferromagnetism of the system.

Let G be a graph on n t> 3 vertices. Dirac's minimum degree condition is the condition that all vertices of G have degree at least in. This is a well-known sufficient condition for the existence of a Hamilton cycle in G. We give related sufficiency conditions for the existence of a Hamilton cycle or a perfect matching involving a restriction of Dirac's minimum degree condition to certain subset...

The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ a finite abelian group order $N$, $A\subseteq G^k$ with $|A|\ge\alpha N^k$. If $M_2$, $M_1-M_2$, $M_1+M_2$ are automorphisms $G^k$, is it true that there exists popular difference $d \in G^k\setminus\{0\}$ such \[\#\...

Large numbers of guanine-rich sequences with potential to form G-quadruplexes have been identified in genomes of various organisms. Such sequences are constrained at both ends by long DNA duplex with a complementary strand in close proximity to compete for duplex formation. G-quadruplex/duplex competition in long double-stranded DNA has rarely been studied. In this work, we used DMS footprintin...