[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

Small Circuit Double Covers of Cubic Multigraphs

A graph is claw-free if it has no induced K1,3 subgraph. A graph is essential 4-edge-connected if removing at most three edges, the resulting graph has at most one component having edges. In this note, we show that every essential 4-edgeconnected claw free graph has a spanning Eulerian subgraph with maximum degree at most 4. The graphs in this paper are finite and loopless. For terms not defined here, see Bondy and Murty [1]. Let G be a graph. Then δ(G) denotes the minimum degree of G. An edge subset X ⊆ E(G) is an essential edge-cut of G if each component of G − X has at least an edge. A graph G is essentially k-edge-connected if for any X ⊆ E(G) with |X| < k, at most one component of G − X has edges. The line graph L(G) of a graph G has E(G) as its vertex set and two vertices of L(G) are adjacent if and only if they are adjacent as edges in G. A graph is called claw-free if it has not induced subgraph isomorphic to K1,3. It is well-known that a line graph is claw-free; and that for a graph G that is not isomorphic to a K1,n−1, G is essentially k-edge-connected if and only if L(G) is k-connected. If G has a cycle containing every vertex of G, then G is called Hamiltonian. A graph G is even if every vertex of G has even degree. A graph is Eulerian if it is connected even graph. A graph with a spanning Eulerian subgraphH with ∆(H) = 2 is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. An Eulerian subgraph H of G is dominating if E(G − V (H)) = ∅. Harary and Nash-Williams [3] showed that for a connected graph G with |E(G)| ≥ 3, L(G) is Hamiltonian if and only if G contains a dominating Eulerian subgraph. The following conjectures are well known. Conjecture (Matthews and Sumner [4]). Every 4-connected claw-free graph is Hamiltonian. Conjecture (Thomassen [6]). Every 4-connected line graphs is Hamiltonian. Ryjacek [5] proved that these two conjectures are equivalent. In this paper, we present a best possible result on claw-free graph with spanning Eulerian subgraphs with maximum degree at most 4. In [2], Catlin introduced the concept of collapsible graphs. A graph G is called collapsible if for every even subset S ⊆ V (G), there is a subgraph H such that G − E(H) is connected and the set of all odd degree vertices in H is equal to S. Note that if a graph G is collapsible, then G is supereulerian. By definition, K1 is both collapsible and supereulerian graph. Catlin showed that every graph G has a unique collection of pairwise vertex-disjoint maximal collapsible subgraphs H1, · · · ,Hc such that ⋃c i=1 V (Hi) = V (G). Let G be a graph and let X ⊆ E(G). The contraction G/X is the graph obtained from G by identifying the ends of each edge in X and deleting the resulting loops. If H is a subgraph of G, then we use G/H for G/E(H). Combining the concepts of collapsible graphs and the contraction, Catlin [2] define the reduction of a graph. The reduction of G, denoted by G′, is the graph obtained from G by contracting each maximal collapsible subgraph Hi, (1 ≤ i ≤ c), into a single vertex vi. A graph G is reduced if G = G′. Each vertex v in G′ defined the maximal collapsible subgraph H in G to be the preimage of v in the contraction. A vertex in G′ is a trivial contraction if its preimage is K1. Theorem A (Catlin [2]). Let G be a graph and let G′ be the reduction of G. Then each of the following holds. (1) G is collapsible if and only if G′ = K1. (2) G is supereulerian if and only if G′ is supereulerian. (3) If G is reduced, then G is K3-free with δ(G) ≤ 3. Theorem 1. Let G be an essential 4-edge-connected claw-free simple graph with δ(G) ≥ 3. Then (1) G is collapsible, and (2) G has spanning Eulerian subgraph H with ∆(H) ≤ 4. Proof. We prove (1) first. By way of contradiction, suppose that G is not collapsible. Let G′ be the reduction of G. By Theorem A (3), δ(G′) ≤ 3. Let v be a vertex in G′ with δ(G′) = dG′(v), and let H be the preimage of v in G. Since δ(G) ≥ 3, and since G is essentially 4-edge-connected, we must have dG′(v) ≥ 3, where equality holds only if |V (H)| = 1. If dG′(v) = 3, then since G is claw-free, two of the three edges incident with v will be in a triangle. It follows that |V (H)| ≥ 3, contrary to the fact that |V (H)| = 1. Therefore, δ(G′) ≥ 4, contrary to Theorem A(3), and so G must be collapsible. Next, we prove (2) of the theorem. Since G is collapsible, G has spanning Eulerian subgraphs. Let H be a spanning Eulerian subgraph of G with the edge set |E(H)| as small as possible. We shall show that ∆(H) ≤ 4. By way of contradiction, there is a vertex v in V (H) such that dH(v) = ∆(H) ≥ 6. Then there are at least 6 vertices adjacent to v in H. Note that since H is Eulerian simple graph, each component of H − v has nonzero even number of vertices adjacent to v in H. Claim 1. Suppose that A = {v1, v2, · · · vr} is the set of vertices in a component of H − v such that each vi is adjacent to v in H, and that r ≥ 4. For any pair of distinct vi, vj ∈ A, if vivj ∈ E(G) then vivj ∈ E(H). If there are i 6= j such that vivj ∈ E(G) − E(H), then since vi and vj with other vertices are in the same component of H − v, H0 = H+vivj −{vvi, vvj} is connected and spanning even subgraph of G. Thus, H0 is a spanning Eulerian subgraph of G with fewer edges than H, a contradiction. Claim 1 holds. Case 1. One component of H − v has at least six vertices adjacent to v in H. Suppose that v1, v2, v3, v4, v5 and v6 are six vertices in a component of H − v such that each vi is adjacent to v in H. Let S = {v, v1, v2, v3, v4, v5, v6}. Let H[S] be the induced subgraph of H by S, and let Ga be the graph obtained from the 5 vertex wheel W4 by removing a rim edge from W4 (see Figure 1). Claim 2. H[S] contains a subgraph isomorphic to K4 or Ga with the given vertices where {a, b, c, d} ⊆ S (Figure 1).

/pdf/spanning-eulerian-subgraphs-in-claw-free-graphs-4pjp3lihjc.pdf

Spanning Eulerian Subgraphs in claw-free graphs

The automatic reconfiguration of electric shipboard power systems is an important step toward improved fight-through and self-healing capabilities of all-electric naval warships. The authors have presented a distributed algorithm for autonomous reconfiguration within an energy management framework using an agent based approach. In the previous work, a blackboard architecture was used as globally shared memory structure for detection of algorithm termination. This paper presents alternative schemes for the detection of global properties, and specifically, detection of algorithm termination.

Distributed Approaches for Determination of Reconfiguration Algorithm Termination

Let denote the class of c-cyclic graphs with n vertices, girth and pendant vertices. In this paper, we determine the unique extremal graph with largest signless Laplacian spectral radius and Laplacian spectral radius in the class of connected c-cyclic graphs with vertices, girth g and at most pendant vertices, respectively, and the unique extremal graph with largest signless Laplacian spectral radius of when and , and we also identify the unique extremal graph with largest Laplacian spectral radius in in the case and either and g is even or and g is odd. Our results extends the corresponding results of [Sci. Sin. Math. 2010;40:1017–1024, Electron. J. Combin. 2011; 18:p.183, Comput. Math. Appl. 2010;59:376–381, Electron. J. Linear Algebra. 2011;22:378–388 and J. Math. Res. Appl. 2014;34:379–391].

The (signless) Laplacian spectral radii of c-cyclic graphs with n vertices, girth g and k pendant vertices

With graphs considered as natural models for many network design problems, edge connectivity ??(G) and maximum number of edge-disjoint spanning trees ?(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G (see Cunningham, in J ACM 32:549---561, 1985; Matula, in Proceedings of 28th Symposium Foundations of Computer Science, pp 249---251, 1987, among others). Mader (Math Ann 191:21---28, 1971) and Matula (J Appl Math 22:459---480, 1972) introduced the maximum subgraph edge connectivity $${\overline{\kappa'}(G) = {\rm max} \{\kappa'(H) : H {\rm is} \, {\rm a} \, {\rm subgraph} \, {\rm of} G \}}$$ ? ? ¯ ( G ) = max { ? ? ( H ) : H is a subgraph of G } . Motivated by their applications in network design and by the established inequalities $$\overline{\kappa'}(G) \ge \kappa'(G) \ge \tau(G),$$ ? ? ¯ ( G ) ? ? ? ( G ) ? ? ( G ) , we present the following in this paper:

Hong-Jian Lai

Papers

Small Circuit Double Covers of Cubic Multigraphs

Spanning Eulerian Subgraphs in claw-free graphs

Distributed Approaches for Determination of Reconfiguration Algorithm Termination

The (signless) Laplacian spectral radii of c-cyclic graphs with n vertices, girth g and k pendant vertices

Characterizations of Strength Extremal Graphs