亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

A $(a,b)$-coloring of a graph $G$ associates to each vertex a $b$-subset of a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoint. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, using necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices, more complex $(a,b)$-colorability reductions are presented. The utility of these tools is exemplified on finite triangle-free induced subgraphs of the triangular lattice for which McDiarmid-Reed's conjecture asserts that they are all $(9,4)$-colorable. Computations on millions of such graphs generated randomly show that our tools allow to find a $(9,4)$-coloring for each of them except for one specific regular shape of graphs (that can be $(9,4)$-colored by an easy ad-hoc process). We thus obtain computational evidence towards the conjecture of McDiarmid\&Reed.

We show that the problem of counting the number of $n$-variable unate functions reduces to the problem of counting the number of $n$-variable monotone functions. Using recently obtained results on $n$-variable monotone functions, we obtain counts of $n$-variable unate functions up to $n=9$. We use an enumeration strategy to obtain the number of $n$-variable balanced monotone functions up to $n=7$. We show that the problem of counting the number of $n$-variable balanced unate functions reduces to the problem of counting the number of $n$-variable balanced monotone functions, and consequently, we obtain the number of $n$-variable balanced unate functions up to $n=7$. Using enumeration, we obtain the numbers of equivalence classes of $n$-variable balanced monotone functions, unate functions and balanced unate functions up to $n=6$. Further, for each of the considered sub-class of $n$-variable monotone and unate functions, we also obtain the corresponding numbers of $n$-variable non-degenerate functions.

Katona and Varga showed that for any rational number $t \in (1/2,1]$, no chordal graph is minimally $t$-tough. We conjecture that no chordal graph is minimally $t$-tough for $t>1/2$ and prove several results supporting the conjecture. In particular, we show that for $t>1/2$, no strongly chordal graph is minimally $t$-tough, no split graph is minimally $t$-tough, and no chordal graph with a universal vertex is minimally $t$-tough.

We establish an asymptotic theory that justifies the third method of removing unwanted variation (RUV-III) theoretically.

A Sturmian word of slope $q$ is the cutting sequence of a half-line $y=qx$. We establish a bijection between sequences of certain prefixes of the Sturmian word of slope $q$, and the $q$-decreasing words, which are binary words whose maximal factors of the form $0^a1^b$ satisfy $q \cdot a > b$ whenever $a>0$. We also show that the number of $q$-decreasing words of length $n$ grows as $\Phi(q)^{n(1 + o(1))}$, where $\Phi(1)$ is the golden ratio, $\Phi(2)$ is equal to the tribonacci constant, and that the function $\Phi(q)$ is strictly increasing, discontinuous at every rational point, and exhibits a nice fractal structure related to the Stern--Brocot tree and Minkowski's question mark function.

Network motifs are recurrent, small-scale patterns of interactions observed frequently in a system. They shed light on the interplay between the topology and the dynamics of complex networks across various domains. In this work, we focus on the problem of counting occurrences of small sub-hypergraph patterns in very large hypergraphs, where higher-order interactions connect arbitrary numbers of system units. We show how directly exploiting higher-order structures speeds up the counting process compared to traditional data mining techniques for exact motif discovery. Moreover, with hyperedge sampling, performance is further improved at the cost of small errors in the estimation of motif frequency. We evaluate our method on several real-world datasets describing face-to-face interactions, co-authorship and human communication. We show that our approximated algorithm allows us to extract higher-order motifs faster and on a larger scale, beyond the computational limits of an exact approach.

We present a comparison study between a cluster and factor graph representation of LDPC codes. In probabilistic graphical models, cluster graphs retain useful dependence between random variables during inference, which are advantageous in terms of computational cost, convergence speed, and accuracy of marginal probabilities. This study investigates these benefits in the context of LDPC codes and shows that a cluster graph representation outperforms the traditional factor graph representation.

The dichromatic number of a digraph is the minimum integer $k$ such that it admits a $k$-dicolouring, i.e. a partition of its vertices into $k$ acyclic subdigraphs. We say that a digraph $D$ is a super-orientation of an undirected graph $G$ if $G$ is the underlying graph of $D$. If $D$ does not contain any pair of symmetric arcs, we just say that $D$ is an orientation of $G$. In this work, we give both lower and upper bounds on the dichromatic number of super-orientations of chordal graphs. We also show a family of orientations of cographs for which the dichromatic number is equal to the clique number of the underlying graph.

We consider the problem of counting 4-cycles ($C_4$) in an undirected graph $G$ of $n$ vertices and $m$ edges (in bipartite graphs, 4-cycles are also often referred to as $\textit{butterflies}$). There have been a number of previous algorithms for this problem based on sorting the graph by degree and using randomized hash tables. These are appealing in theory due to compact storage and fast access on average. But, the performance of hash tables can degrade unpredictably and are also vulnerable to adversarial input. We develop a new simpler algorithm for counting $C_4$ requiring $O(m\bar\delta(G))$ time and $O(n)$ space, where $\bar \delta(G) \leq O(\sqrt{m})$ is the $\textit{average degeneracy}$ parameter introduced by Burkhardt, Faber \& Harris (2020). It has several practical improvements over previous algorithms; for example, it is fully deterministic, does not require any sorting of the input graph, and uses only addition and array access in its inner loops. To the best of our knowledge, all previous efficient algorithms for $C_4$ counting have required $\Omega(m)$ space in addition to storing the input graph. Our algorithm is very simple and easily adapted to count 4-cycles incident to each vertex and edge. Empirical tests demonstrate that our array-based approach is $4\times$ -- $7\times$ faster on average compared to popular hash table implementations.

We consider the two-pronged fork frame $F$ and the variety $\mathbf{Eq}(B_F)$ generated by its dual closure algebra $B_F$. We describe the finite projective algebras in $\mathbf{Eq}(B_F)$ and give a purely semantic proof that unification in $\mathbf{Eq}(B_F)$ is finitary and not unitary.