Minimal Euclidean representations of graphs

A simple graph G is representable in a real vector space of dimension m, if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values, α and β, with distance α if the vertices are adjacent and distance β otherwise. The Euclidean representation number of G is the smallest dimension in which G is representable. In this note, we bound the Euclidean representation number of a graph using multiplicities of the eigenvalues of the adjacency matrix. We also give an exact formula for the Euclidean representation number using the main angles of the graph.     

双线性型图与交错型图不是完美图

本文证明了双线性型图与交错型图都不是完美图,从而解决了双线性型图与交错型图的完美图判别问题.     

关于二次型图的完美性

本文应用强完美图定理,解决了二次型图的完美图判别问题.     

A new closure concept preserving graph Hamiltonicity and based on neighborhood equivalence

A graph is Hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is Hamiltonian is known as an NP-complete problem, and no satisfactory characterization for these graphs has been found.In 1976, Bondy and Chvàtal introduced a way to get round the Hamiltonicity problem complexity by using a closure of the graph. This closure is a supergraph of G which is Hamiltonian iff G is. In particular, if the closure is the complete graph, then G is Hamiltonian. Since this seminal work, several closure concepts preserving Hamiltonicity have been introduced. In particular, in 1997, Ryjá?ek defined a closure concept for claw-free graphs based on local completion.Following a different approach, in 1974, Goodman and Hedetniemi gave a sufficient condition for Hamiltonicity based on the existence of a clique covering of the graph. This condition was recently generalized using the notion of Eulerian clique covering. In this context, closure concepts based on local completion are interesting since the closure of a graph contains more simplicial vertices than the graph itself, making the search for a clique covering easier.In this article, we introduce a new closure concept based on local completion which preserves the Hamiltonicity for every graph. Note that, moreover, the closure may be claw free even when the graph is not.     

Strong Embeddability for Groups Acting on Metric Spaces

The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, the permanence properties of strong embeddability for groups acting on metric spaces are studied. The authors show that a finitely generated group acting on a finitely asymptotic dimension metric space by isometries whose $K$-stabilizers are strongly embeddable is strongly embeddable. Moreover, they prove that the fundamental group of a graph of groups with strongly embeddable vertex groups is also strongly embeddable.     

The relationship between the threshold dimension of split graphs and various dimensional parameters

Let the coboxicity of a graph G be denoted by cob(G), and the threshold dimension by t(G). For fixed k≥3, determining if cob(G)≥k and t(G)≤k are both NP-complete problems. We show that if G is a comparability graph, then we can determine if cob(G)≤2 in polynomial time. This result shows that it is possible to determine if the interval dimension of a poset equals 2 in polynomial time. If the clique covering number of G is 2, we show that one can determine if t(G)≤2 in polynomial time. Sufficient conditions on G are given for cob(G)≤2 and for t(G)≤2.     

The neighborhood complex of a random graph

For a graph G, the neighborhood complex N[G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well-known result of Lovász that if ‖N[G]‖ is k-connected, then the chromatic number of G is at least k+3.We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, O(logd), compared to the expected dimension d of the complex itself.     

最小填充问题的可分解性

起源于稀疏矩阵计算和其它应用领域的图G的最小填充问题是在图G中寻求一个内含边数最小的边集F使得G F是弦图.这里最小值|F|称为图G的填充数,表示为f(G).作为NP-困难问题,该问题的降维性质已被研究,其中包括它的可分解性.基本的可分解定理是:如果图G的一个点割集S是一个团,则G经由S是可分解的.作为推广,如果S是一个"近似"团(即只有极少数边丢失的团),则G经由S是可分解的.本文首先给出基本分解定理的另外一个推广:如果S是G的一个极小点割集且G-S含有至少|S|个分支,则G经由S是可分解的;其次,给出了这个新推广定理的一些应用.     

A note on immersion minors and planarity

Graph minors play an important role in graph theory. The focus of this paper is on immersion minors and their relationship to planarity. In general, planar graphs can have non-planar immersion minors. This paper shows that by placing a simple restriction on the immersion-minor operations, all immersion minors of a planar graph are planar. This then allows one to easily obtain a characterization of planar graphs using immersion minors. A dual form of this characterization, as well as an extension to binary matroids, are also considered.     

On the Hadwiger's conjecture for graph products

The Hadwiger number η(G) of a graph G is the largest integer h such that the complete graph on h nodes K_h is a minor of G. Equivalently, η(G) is the largest integer such that any graph on at most η(G) nodes is a minor of G. The Hadwiger's conjecture states that for any graph G, η(G)?χ(G), where χ(G) is the chromatic number of G. It is well-known that for any connected undirected graph G, there exists a unique prime factorization with respect to Cartesian graph products. If the unique prime factorization of G is given as G₁□G₂□?□G_k, where each G_i is prime, then we say that the product dimension of G is k. Such a factorization can be computed efficiently.In this paper, we study the Hadwiger's conjecture for graphs in terms of their prime factorization. We show that the Hadwiger's conjecture is true for a graph G if the product dimension of G is at least . In fact, it is enough for G to have a connected graph M as a minor whose product dimension is at least , for G to satisfy the Hadwiger's conjecture. We show also that if a graph G is isomorphic to F^d for some F, then η(G)?χ(G)^{⌊(d-1)/2⌋}, and thus G satisfies the Hadwiger's conjecture when d?3. For sufficiently large d, our lower bound is exponentially higher than what is implied by the Hadwiger's conjecture.Our approach also yields (almost) sharp lower bounds for the Hadwiger number of well-known graph products like d-dimensional hypercubes, Hamming graphs and the d-dimensional grids. In particular, we show that for the d-dimensional hypercube H_d, . We also derive similar bounds for G^d for almost all G with n nodes and at least edges.     

Graph pegging numbers

In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like “pegging moves” allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging number (respectively, the optimal pegging number) of a graph is the minimum number of pegs such that for every (respectively, some) distribution of that many pegs on the graph, any vertex can be reached by a sequence of pegging moves. We prove several basic properties of pegging and analyze the pegging number and optimal pegging number of several classes of graphs, including paths, cycles, products with complete graphs, hypercubes, and graphs of small diameter.     

A class of highly symmetric graphs,symmetric cylindrical constructions and their spectra

In this article, we introduce the algebra of block-symmetric cylinders and we show that symmetric cylindrical constructions on base-graphs admitting commutative decompositions behave as generalized tensor products. We compute the characteristic polynomial of such symmetric cylindrical constructions in terms of the spectra of the base-graph and the cylinders in a general setting. This gives rise to a simultaneous generalization of some well-known results on the spectra of a variety of graph amalgams, as various graph products, graph subdivisions and generalized Petersen graph constructions. While our main result introduces a connection between spectral graph theory and commutative decompositions of graphs, we focus on commutative cyclic decompositions of complete graphs and tree-cylinders along with a subtle group labeling of trees to introduce a class of highly symmetric graphs containing the Petersen and the Coxeter graphs. Also, using techniques based on recursive polynomials we compute the characteristic polynomials of these highly symmetric graphs as an application of our main result.     

Four Constructions of Highly Symmetric Tetravalent Graphs

Given a connected, dart‐transitive, cubic graph, constructions of its Hexagonal Capping and its Dart Graph are considered. In each case, the result is a tetravalent graph which inherits symmetry from the original graph and is a covering of the line graph.Similar constructions are then applied to a map (a cellular embedding of a graph in a surface) giving tetravalent coverings of the medial graph. For each construction, conditions on the graph or the map to make the constructed graph dart‐transitive, semisymmetric or ‐transitive are considered.     

Closest 4-leaf power is fixed-parameter tractable

The NP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on Closest 2-Leaf Power and Closest 3-Leaf Power, we give the first algorithmic result for Closest 4-Leaf Power, showing that Closest 4-Leaf Power is fixed-parameter tractable with respect to the parameter r.     

Rank of maximum matchings in a graph

The matching polytope is the convex hull of the incidence vectors of all (not necessarily perfect) matchings of a graphG. We consider here the problem of computing the dimension of the face of this polytope which contains the maximum cardinality matchings ofG and give a good characterization of this quantity, in terms of the cyclomatic number of the graph and families of odd subsets of the nodes which are always nearly perfectly matched by every maximum matching.This is equivalent to finding a maximum number of linearly independent representative vectors of maximum matchings ofG; the size of such a set is called thematching rank ofG. We also give in the last section a way of computing that rank independently of those parameters.Note that this gives us a good lower bound on the number of those matchings.     

The spum and sum-diameter of graphs: Labelings of sum graphs

A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph G is the minimum difference between the largest and smallest labels in a sum graph consisting of G and the minimum number of additional isolated vertices necessary so that a sum graph labeling exists. We investigate the spum of various families of graphs, namely cycles, paths, and matchings. We introduce the sum-diameter, a modification of the definition of spum that omits the requirement that the number of additional isolated vertices in the sum graph is minimal, which we believe is a more natural quantity to study. We then provide asymptotically tight general bounds on both sides for the sum-diameter, and study its behavior under numerous binary graph operations as well as vertex and edge operations. Finally, we generalize the sum-diameter to hypergraphs.     

Topological graph dimension

In the invited chapter Discrete Spatial Models of the book Handbook of Spatial Logics, we have introduced the concept of dimension for graphs, which is inspired by Evako’s idea of dimension of graphs [A.V. Evako, R. Kopperman, Y.V. Mukhin, Dimensional properties of graphs and digital spaces, J. Math. Imaging Vision 6 (1996) 109-119]. Our definition is analogous to that of (small inductive) dimension in topology. Besides the expected properties of isomorphism-invariance and monotonicity with respect to subgraph inclusion, it has the following distinctive features:

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Local aspect. That is, dimension at a vertex is basic, and the dimension of a graph is obtained as the sup over its vertices.

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Dimension of a strong product G×H is dim(G)+dim(H) (for non-empty graphs G,H).

In this paper we present a short account of the basic theory, with several new applications and results.     

Solving some NP-complete problems using split decomposition

We show how to use the split decomposition to solve some NP-hard optimization problems on graphs. We give algorithms for clique problem and domination-type problems. Our main result is an algorithm to compute a coloration of a graph using its split decomposition. Finally we show that the clique-width of a graph is bounded if and only if the clique-width of each representative graph in its split decomposition is bounded.     

On the Laplacian Eigenvalues of Signed Graphs

A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the largest Laplacian eigenvalue of a signed graph is investigated, which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.     

Rectangular drawings of plane graphs without designated corners

This paper addresses the problem of finding rectangular drawings of plane graphs, in which each vertex is drawn as a point, each edge is drawn as a horizontal or a vertical line segment, and the contour of each face is drawn as a rectangle. A graph is a 2–3 plane graph if it is a plane graph and each vertex has degree 3 except the vertices on the outer face which have degree 2 or 3. A necessary and sufficient condition for the existence of a rectangular drawing has been known only for the case where exactly four vertices of degree 2 on the outer face are designated as corners in a 2–3 plane graph G. In this paper we establish a necessary and sufficient condition for the existence of a rectangular drawing of G for the general case in which no vertices are designated as corners. We also give a linear-time algorithm to find a rectangular drawing of G if it exists.