Polar graphs and maximal independent sets

We study two central problems of algorithmic graph theory: finding maximum and minimum maximal independent sets. Both problems are known to be NP-hard in general. Moreover, they remain NP-hard in many special classes of graphs. For instance, the problem of finding minimum maximal independent sets has been recently proven to be NP-hard in the class of so-called (1,2)-polar graphs. On the other hand, both problems can be solved in polynomial time for (1,1)-polar, also known as split graphs. In this paper, we address the question of distinguishing new classes of graphs admitting polynomial-time solutions for the two problems in question. To this end, we extend the hierarchy of (α,β)-polar graphs and study the computational complexity of the problems on polar graphs of special types.     

Path coupling using stopping times and counting independent sets and colorings in hypergraphs

We analyse the mixing time of Markov chains using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Δ of a vertex and the minimum size m of an edge satisfy m ≥ 2Δ + 1. We also show that the Glauber dynamics for proper q‐colorings of a hypergraph mixes rapidly if m ≥ 4 and q > Δ, and if m = 3 and q ≥ 1.65Δ. We give related results on the hardness of exact and approximate counting for both problems. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Estimates of the number of independent sets in graphs with a fixed independence number

Attainable estimates of the number of independent sets in graphs with a given size of the maximal independent set are obtained. Three graph classes—trees, forests, and the class of all graphs—are considered. Extremal graphs are described.     

Graphs with the second largest number of maximal independent sets

Let G be a simple undirected graph. Denote by (respectively, xi(G)) the number of maximal (respectively, maximum) independent sets in G. Erd?s and Moser raised the problem of determining the maximum value of among all graphs of order n and the extremal graphs achieving this maximum value. This problem was solved by Moon and Moser. Then it was studied for many special classes of graphs, including trees, forests, bipartite graphs, connected graphs, (connected) triangle-free graphs, (connected) graphs with at most one cycle, and recently, (connected) graphs with at most r cycles. In this paper we determine the second largest value of and xi(G) among all graphs of order n. Moreover, the extremal graphs achieving these values are also determined.     

On the number of minimal transversals in 3-uniform hypergraphs

We prove that the number of minimal transversals (and also the number of maximal independent sets) in a 3-uniform hypergraph with n vertices is at most cⁿ, where c≈1.6702. The best known lower bound for this number, due to Tomescu, is adⁿ, where d=10^1/5≈1.5849 and a is a constant.     

Trees with the second largest number of maximal independent sets

A maximal independent set is an independent set that is not a proper subset of any other independent set. In this paper, we determine the second largest number of maximal independent sets among all trees and forests of order n≥4. We also characterize those extremal graphs achieving these values.     

The maximum number of maximal independent sets in unicyclic connected graphs

In this paper, we determine the maximum number of maximal independent sets in a unicyclic connected graph. We also find a class of graphs achieving this maximum value.     

Maximal and maximum independent sets in graphs with at most r cycles

Let m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a formula for c(n,r) when r is large relative to n, while a theorem of Goh, Koh, Sagan, and Vatter does the same when r is small relative to n. We complete the determination of c(n,r) for all n and r and characterize the extremal graphs. Problems for maximum independent sets are also completely resolved. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 283–314, 2006     

Constraints on the number of maximal independent sets in graphs

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3^n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.     

Randomized on-line algorithms and lower bounds for computing large independent sets in disk graphs

We study the on-line version of the maximum independent set problem, for the case of disk graphs which are graphs resulting from intersections of disks on the plane. In particular, we investigate whether randomization can be used to break known lower bounds for deterministic on-line independent set algorithms and present new upper and lower bounds.     

Counting the number of independent sets in chordal graphs

We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant time per output. On the other hand, we prove that the following problems for a chordal graph are #P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is NP-hard, and even hard to approximate.     

Independent sets of maximal size in tensor powers of vertex‐transitive graphs

Let G be a connected, nonbipartite vertex‐transitive graph. We prove that if the only independent sets of maximal cardinality in the tensor product G × G are the preimages of the independent sets of maximal cardinality in G under projections, then the same holds for all finite tensor powers of G, thus providing an affirmative answer to a question raised by Larose and Tardif (J Graph Theory 40(3) (2002), 162–171). © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 295‐301, 2009     

关于向量组的秩与最大无关组的教学改革研究

对向量的秩与最大无关组的教学过程进行了分析研究，从教学内容组织与教学手段等方面提出了改革的方案．对有的教材的不足提出了改进办法.     

Primitivity and independent sets in direct products of vertex‐transitive graphs

We introduce the concept of the primitivity of independent set in vertex‐transitive graphs, and investigate the relationship between the primitivity and the structure of maximum independent sets in direct products of vertex‐transitive graphs. As a consequence of our main results, we positively solve an open problem related to the structure of independent sets in powers of vertex‐transitive graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 218‐225, 2011     

Invariant Gaussian processes and independent sets on regular graphs of large girth

We prove that every 3‐regular, n‐vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d‐regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue . We show that such processes can be approximated by i.i.d. factors provided that . We then use these approximations for to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 284–303, 2015     

格中依次最短无关组与Minkowski约化基等价的充分条件

本文讨论了格中基子集、依次最短无关组及Minkowski约化基之间的向量长度关系,利用无关组与基之间的一些制约性质,给出了Minkowski约化基达到依次最短长度,以及依次最短无关组成为Minkowski约化基的一些充分条件.     

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An interval$t$-coloring of a multigraph $G$ is a proper edge coloring with colors $1,\dots ,t$ such that the colors of the edges incident with every vertex of $G$ are colored by consecutive colors. A cyclic interval$t$-coloring of a multigraph $G$ is a proper edge coloring with colors $1,\dots ,t$ such that the colors of the edges incident with every vertex of $G$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. Denote by $w\left(G\right)$ ($w_c\left(G\right)$) and $W\left(G\right)$ ($W_c\left(G\right)$) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph $G$, respectively. We present some new sharp bounds on $w\left(G\right)$ and $W\left(G\right)$ for multigraphs $G$ satisfying various conditions. In particular, we show that if $G$ is a $2$-connected multigraph with an interval coloring, then $W\left(G\right)\le 1+?\frac{\left|V\left(G\right)\right|}{2}?(\Delta \left(G\right)?1)$. We also give several results towards the general conjecture that $W_c\left(G\right)\le \left|V\left(G\right)\right|$ for any triangle-free graph $G$ with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most $4$.     

乘积FC-空间内涉及——较好容许集值映象的优化映象族的极大元及其应用

引入了涉及-较好容许集值映象的映-拓扑空间到-有限连续拓扑空间（简称，FC-空间）的优化映象族．在乘积FC-空间的非紧设置下埘这类优化映象族证明了某些极大元存在性定理．在乘积FC-空间内给出了对不动点和极小极大不等式组的应用．这些定理改进、统一和推广了最近文献中的很多重要结果．     

Functions on discrete sets holomorphic in the sense of Ferrand, or monodiffric functions of the second kind

We study the class of functions called monodiffric of the second kind by Isaac.They are discrete analogues of holomorphic functions of one or two complex variables.Discrete analogues of the Cauchy-Riemann operator,of domains of holomorphy in one discrete variable,and of the Hartogs phenomenon in two discrete variables are investigated.Two fundamental solutions to the discrete Cauchy-Riemann equation are studied:one with support in a quadrant,the other with decay at infinity.The first is easy to construct by induction;the second is accessed via its Fourier transform.