Let $Q_d$ be the $d$-dimensional Hamming cube and $N=|V(Q_d)|=2^d$. An independent set $I$ in is called balanced if contains same number of even odd vertices. We show that logarithm sets is
 \[(1-\Theta(1/\sqrt d))N/2.\]
 The key ingredient proof an improved version "Sapozhenko's graph container lemma".

Let $G=(V,E)$ be a simple graph. A set $Ssubseteq V$ isindependent set of $G$,  if no two vertices of $S$ are adjacent.The  independence number $alpha(G)$ is the size of a maximumindependent set in the graph. In this paper we study and characterize the independent sets ofthe zero-divisor graph $Gamma(R)$ and ideal-based zero-divisor graph $Gamma_I(R)$of a commutative ring $R$.

Let r be an integer, f(n) a function, and H a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi [8], the r-Ramsey-Turán number of H , RTr(n,H, f(n)), is defined to be the maximum number of edges in an n-vertex, H-free graph G with αr(G) ≤ f(n) where αr(G) denotes the Kr-independence number of G. In this note, using isoperimetric properties of the high dimensional unit sphere, we construct ...

A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V,E), a subset D ⊆ V (G) is a 2dominating set if every vertex of V (...

The Turán connected graph TCn,α is obtained from α cliques of size b αc or d αe by joining all cliques by an edge to one central vertex in one of the larger cliques. The graph TCn,α was shown recently by Bruyère and Mélot to maximise the number of independent sets among connected graphs of order n and independence number α. We prove a generalisation of this result by showing that TCn,α in fact ...

A dominating set in a graph G is a set S of vertices such that every vertex outside S has a neighbor in S; the domination number γ(G) is the minimum size of such a set. The independent domination number, written i(G), is the minimum size of a dominating set that also induces no edges. Henning and Southey conjectured that if G is a connected cubic graph with sufficiently many vertices, then i(G)...

We let γ(G) and i(G) denote the domination number and the independent domination number ofG, respectively. Recently, Rad and Volkmann conjectured that i(G)/γ(G) ≤ ∆(G)/2 for every graph G, where ∆(G) is the maximum degree of G. In this note, we construct counterexamples of the conjecture for ∆(G) ≥ 6, and give a sharp upper bound of the ratio i(G)/γ(G) by using the maximum degree of G.

MOTIVATION Somatic copy number aberrations SCNAS: are frequent in cancer genomes, but many of these are random, passenger events. A common strategy to distinguish functional aberrations from passengers is to identify those aberrations that are recurrent across multiple samples. However, the extensive variability in the length and position of SCNA: s makes the problem of identifying recurrent ab...

Let G be a simple undirected graph. Denote by mi(G) (respectively, xi(G)) the number of maximal (respectively, maximum) independent sets in G. In this paper we determine the third and fouth largest value of mi(G) among all graphs of order n. Moreover, the extremal graphs achieving these values are also determined. Mathematics Subject Classification: 05C35, 05C69, 68R10

We study the problem of estimating the number of independent motions for segmentation based on feature tracking. We elucidate the mathematical structure of the problem and present an estimation method using model selection by the geometric AIC, the geometric MDL, and the OIC. We compare their performance using synthetic and real data. We also present techniques for evaluating the reliability of...