[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

Graphs without spanning closed trails

Let A be a finite abelian group and G be a digraph. The boundary of a func- tion f : EOGU7!A is a function q f : VOGU7!A given by q fOvUa P e leaving v fOeUˇ P e entering v fOeU. The graph G is A-connected if for every b : VOGU7!A with P vA VOGU bOvUa0, there is a function f : EOGU7!Af0g such that q f a b. In (J. Combi- natorial Theory, Ser. B 56 (1992) 165-182), Jaeger et al showed that every 3-edge- connected graph is A-connected, for every abelian group A with jAjV 6. It is con- jectured that every 3-edge-connected graph is A-connected, for every abelian group A with jAjV 5; and that every 5-edge-connected graph is A-connected, for every abelian group A with jAjV 3. In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A withjAjV 3.

Group Connectivity of 3-Edge-Connected Chordal Graphs

Hypercube is one of the most popular topologies for connecting processors in multicomputer systems. In this paper we address the maximum order of a connected component in a faulty cube. The results established include several known conclusions as special cases. We conclude that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.

On the maximal connected component of hypercube with faulty vertices

A proper vertex k-coloring of a graph G is dynamic if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic k-coloring is the dynamic chromatic number χd(G). We prove in this paper the following best possible upper bounds as an analogue to the Brook’s Theorem, together with the determination of chromatic numbers for complete k-partite graphs. (1) If ∆ ≤ 3, then χd(G) ≤ 4, with the only exception that G = C5, in which case χd(C5) = 5. (2) If ∆ ≥ 4, then χd(G) ≤ ∆+ 1. (3) χd(K1,1) = 2, χd(K1,m) = 3 and χd(Km,n) = 4 for m,n ≥ 2; χd(Kn1,n2,···,nk) = k for k ≥ 3.

Hong-Jian Lai

Papers

Graphs without spanning closed trails

Group Connectivity of 3-Edge-Connected Chordal Graphs

On the maximal connected component of hypercube with faulty vertices

Upper Bounds of Dynamic Chromatic Number.

Fractional arboricity, strength, and principal partitions in graphs and matroids