[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Swarm robotics is an approach to collective robotics that takes inspiration from the self-organized behaviors of social animals. Through simple rules and local interactions, swarm robotics aims at designing robust, scalable, and flexible collective behaviors for the coordination of large numbers of robots. In this paper, we analyze the literature from the point of view of swarm engineering: we focus mainly on ideas and concepts that contribute to the advancement of swarm robotics as an engineering field and that could be relevant to tackle real-world applications. Swarm engineering is an emerging discipline that aims at defining systematic and well founded procedures for modeling, designing, realizing, verifying, validating, operating, and maintaining a swarm robotics system. We propose two taxonomies: in the first taxonomy, we classify works that deal with design and analysis methods; in the second taxonomy, we classify works according to the collective behavior studied. We conclude with a discussion of the current limits of swarm robotics as an engineering discipline and with suggestions for future research directions.

/pdf/swarm-robotics-a-review-from-the-swarm-engineering-46mfc058e6.pdf

Swarm robotics: a review from the swarm engineering perspective

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

/pdf/graph-coloring-problems-4v7vijos3s.pdf

Graph Coloring Problems

: Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization. There has also been research on the application of techniques from combinational optimization to nonlinear problems.

http://www.dtic.mil/dtic/tr/fulltext/u2/a273552.pdf

Discrete Applied Mathematics

Let $$H=(V,E)$$
 be a hypergraph, where V is a set of vertices and E is a set of non-empty subsets of V called edges. If all edges of H have the same cardinality r, then H is a r-uniform hypergraph; if E consists of all r-subsets of V, then H is a complete r-uniform hypergraph, denoted by $$K_n^r$$
 , where $$n=|V|$$
 . A hypergraph $$H'=(V',E')$$
 is called a subhypergraph of $$H=(V,E)$$
 if $$V'\subseteq V$$
 and $$E'\subseteq E$$
 . A r-uniform hypergraph $$H=(V,E)$$
 is vertex-k-maximal if every subhypergraph of H has vertex-connectivity at most k, but for any edge $$e\in E(K_n^r)\setminus E(H)$$
 , $$H+e$$
 contains at least one subhypergraph with vertex-connectivity at least $$k+1$$
 . In this paper, we first prove that for given integers n, k, r with $$k,r\ge 2$$
 and $$n\ge k+1$$
 , every vertex-k-maximal r-uniform hypergraph H of order n satisfies $$|E(H)|\ge (^n_r)-(^{n-k}_r)$$
 , and this lower bound is best possible. Next, we conjecture that for sufficiently large n, every vertex-k-maximal r-uniform hypergraph H on n vertices satisfies $$|E(H)|\le (^n_r)-(^{n-k}_r)+(\frac{n}{k}-2)(^k_r)$$
 , where $$k,r\ge 2$$
 are integers. And the conjecture is verified for the case $$r>k$$
 .

On the Sizes of Vertex-k-Maximal r-Uniform Hypergraphs

Characterization of minimally (2,l)-connected graphs

List r-hued chromatic number of graphs with bounded maximum average degrees

Realizing degree sequences as Z3 -connected graphs

Let G be a simple graph with n vertices and let G c denote the complement of G . Let ω(G) denote the number of components of G and G ( E ) the spanning subgraph of G with edge set E . Suppose that | E |= k (| V ( G ) |-1). Consider the partition Π=〈 X 1 ,  X 2 , . . . ,  X k 〉 of E ( G ) such that | X i |= n -1 (1≤ i ≤ k ). Define ϵ( Π)=Σ k i=1  ω(G(X i ))-k and ϵ(G)= min  ϵ(Π), where the minimum is taken over all such partitions. In [ Europ. J. Combin. 7 (1986), 263-270], Payan conjectures that if ϵ(G) >0, then there exist edges e ∈ E ( G ) and e′∈E(G c ) such that ϵ ( G - e + e ′) G ). This conjecture will be proved in this note.

Hong-Jian Lai

Papers

On the Sizes of Vertex-k-Maximal r-Uniform Hypergraphs

Characterization of minimally (2,l)-connected graphs

List r-hued chromatic number of graphs with bounded maximum average degrees

Realizing degree sequences as Z3 -connected graphs

A property on edge-disjoint spanning trees