Unavoidable Configurations in Complete Topological Graphs

A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least clog^1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most clog^1/6 n vertices.     

Greedy maximum-clique decompositions

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. We have recently shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. A greedy max-clique decomposition is a particular kind cf greedy clique decomposition where maximum cliques are removed, instead of just maximal ones. In this paper, we show that any greedy max-clique decompositionC of a graph of ordern has, wheren(C) is the number of vertices inC.     

On clique convergent graphs

A graphG isconvergent when there is some finite integern 0, such that then-th iterated clique graphK ⁿ(G) has only one vertex. The smallest suchn is theindex ofG. TheHelly defect of a convergent graph is the smallestn such thatK ⁿ(G) is clique Helly, that is, its maximal cliques satisfy the Helly property. Bandelt and Prisner proved that the Helly defect of a chordal graph is at most one and asked whether there is a graph whose Helly defect exceeds the difference of its index and diameter by more than one. In the present paper an affirmative answer to the question is given. For any arbitrary finite integern, a graph is exhibited, the Helly defect of which exceeds byn the difference of its index and diameter.     

The n‐ordered graphs: A new graph class

For a positive integer n, we introduce the new graph class of n‐ordered graphs, which generalize partial n‐trees. Several characterizations are given for the finite n‐ordered graphs, including one via a combinatorial game. We introduce new countably infinite graphs R⁽ⁿ⁾, which we name the infinite random n‐ordered graphs. The graphs R⁽ⁿ⁾ play a crucial role in the theory of n‐ordered graphs, and are inspired by recent research on the web graph and the infinite random graph. We characterize R⁽ⁿ⁾ as a limit of a random process, and via an adjacency property and a certain folding operation. We prove that the induced subgraphs of R⁽ⁿ⁾ are exactly the countable n‐ordered graphs. We show that all countable groups embed in the automorphism group of R⁽ⁿ⁾. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 204–218, 2009     

Vertex-transitive graphs that are not Cayley graphs. II

The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph that is not a Cayley graph. In 1983, D. Marus˘ic˘ asked, “For what values of n does there exist such a graph on n vertices?” We give several new constructions of families of vertex-transitive graphs that are not Cayley graphs and complete the proof that, if n is divisible by p² for some prime p, then there is a vertex-transitive graph on n vertices that is not a Cayley graph unless n is p², p³, or 12. © 1996 John Wiley & Sons, Inc.     

Graphs with exactly one hamiltonian circuit

Let h(n) be the largest integer such that there exists a graph with n vertices having exactly one Hamiltonian circuit and exactly h(n) edges. We prove that h(n) = [n²/4]+1 (n ≧ 4) and discuss some related problems.     

An algorithm for the enumeration of spanning trees

Enumeration of spanning trees of an undirected graph is one of the graph problems that has received much attention in the literature. In this paper a new enumeration algorithm based on the idea of contractions of the graph is presented. The worst-case time complexity of the algorithm isO(n+m+nt) wheren is the number of vertices,m the number of edges, andt the number of spanning trees in the graph. The worst-case space complexity of the algorithm isO(n ²). Computational analysis indicates that the algorithm requires less computation time than any other of the previously best-known algorithms.     

Extremal subgraphs of random graphs

We shall prove that if L is a 3-chromatic (so called “forbidden”) graph, and —Rⁿ is a random graph on n vertices, whose edges are chosen independently, with probability p, and —Bⁿ is a bipartite subgraph of Rⁿ of maximum size, —Fⁿ is an L-free subgraph of Rⁿ of maximum size, then (in some sense) Fⁿ and Bⁿ are very near to each other: almost surely they have almost the same number of edges, and one can delete O_p(1) edges from Fⁿ to obtain a bipartite graph. Moreover, with p = 1/2 and L any odd cycle, Fⁿ is almost surely bipartite.     

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism.We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2^n−pn^O(1) or in time 3^n−pn^O(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.     

The random planar graph process

We consider the following variant of the classical random graph process introduced by Erd?s and Rényi. Starting with an empty graph on n vertices, choose the next edge uniformly at random among all edges not yet considered, but only insert it if the graph remains planar. We show that for all ε > 0, with high probability, θ(n²) edges have to be tested before the number of edges in the graph reaches (1 + ε)n. At this point, the graph is connected with high probability and contains a linear number of induced copies of any fixed connected planar graph, the first property being in contrast and the second one in accordance with the uniform random planar graph model. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Competition graphs and clique dimensions

Here it is proved that for almost all simple graphs over n vertices one needs Ω(n^4/3(log n)^?4/3) extra vertices to obtain them as a double competition graph of a digraph. on the other hand O(n^5/3) extra vertices are always sufficient. Several problems remain open.     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Contact patterns of equal nonoverlapping spheres

LetF be a set of nonoverlapping spheres in Euclideann-spaceE ⁿ . By the contact pattern ofF we mean the graph whose vertex set isF and two vertices are adjacent whenever the corresponding spheres touch each other. Every graph turns out to be a contact pattern in some dimension. This paper studies the smallest dimensionn for a graphG such thatG is a contact pattern inE ⁿ . Among others, the smallest dimensions are determined for the join of a large complete graph and an empty graph, and for complete multipartite graphs with more vertex classes than the size of its largest vertex class.     

Note on induced subgraphs of the unit distance graphE n

The unit distance graphE ⁿ is the graph whose vertices are the points in Euclideann-space, and two vertices are adjacent if and only if the distance between them is 1. We prove that for anyn there is a finite bipartite graph which cannot be embedded inE ⁿ as an induced subgraph and that every finite graph with maximum degreed can be embedded inE ^N ,N=(d ³ –d)/2, as an induced subgraph.     

On graphs preserving rectilinear shortest paths in the presence of obstacles

In physical VLSI design, network design (wiring) is the most time-consuming phase. For solving global wiring problems, we propose to first compute from the layout geometry a graph that preserves all shortest paths between pairs of relevant points, and then to operate on that graph for computing shortest paths, Steiner minimal tree approximations, or the like. For a set of points and a set of simple orthogonal polygons as obstacles in the plane, withn input points (polygon corner or other) altogether, we show how a shortest paths preserving graph of sizeO(n logn) can be computed in timeO(n logn) in the worst case, with spaceO(n). We illustrate the merits of this approach with a simple example: If the length of a longest edge in the graph is bounded by a polynomial inn, an assumption that is clearly fulfilled for graphs derived from VLSI layout geometries, then a shortest path can be computed in timeO(n logn log logn) in the worst case; this result improves on the known best one ofO(n(logn)^3/2).     

Diameters of iterated clique graphs of chordal graphs

The clique graph K(G) of a graph is the intersection graph of maximal cliques of G. The iterated clique graph Kⁿ(G) is inductively defined as K(K^n?1(G)) and K¹(G) = K(G). Let the diameter diam(G) be the greatest distance between all pairs of vertices of G. We show that diam(Kⁿ(G)) = diam(G) — n if G is a connected chordal graph and n ≤ diam(G). This generalizes a similar result for time graphs by Bruce Hedman.     

Phase transition of the minimum degree random multigraph process

We study the phase transition of the minimum degree multigraph process. We prove that for a constant h_g ≈︁ 0.8607, with probability tending to 1 as n → ∞, the graph consists of small components on O(log n) vertices when the number of edges of a graph generated so far is smaller than h_gn, the largest component has order roughly n^2/3 when the number of edges added is exactly h_gn, and the graph consists of one giant component on Θ(n) vertices and small components on O(log n) vertices when the number of edges added is larger than h_gn. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Fast parallel graph searching with applications

This paper presents fast parallel algorithms for the following graph theoretic problems: breadth-depth search of directed acyclic graphs; minimum-depth search of graphs; finding the minimum-weighted paths between all node-pairs of a weighted graph and the critical activities of an activity-on-edge network. The first algorithm hasO(logdlogn) time complexity withO(n ³) processors and the remaining algorithms achieveO(logd loglogn) time bound withO(n ²[n/loglogn]) processors, whered is the diameter of the graph or the directed acyclic graph (which also represents an activity-on-edge network) withn nodes. These algorithms work on an unbounded shared memory model of the single instruction stream, multiple data stream computer that allows both read and write conflicts.     

A strongly regularN-full graph of small order

For every positive integern we show the construction of a strongly regular graph of order at most 2 ⁿ⁺² which contains every graph of ordern as a subgraph. The estimation concerning the construction is best possible.     

Erratum to: On the Extremal Zagreb Indices of Graphs with Cut Edges

For a (molecular) graph, the first Zagreb index M ₁ is equal to the sum of squares of the vertex degrees, and the second Zagreb index M ₂ is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we show that all connected graphs with n vertices and k cut edges, the maximum (resp. minimum) M ₁- and M ₂-value are obtained, respectively, and uniquely, at K _n ^k (resp. P _n ^k ), where K _n ^k is a graph obtained by joining k independent vertices to one vertex of K _n−k and P _n ^k is a graph obtained by connecting a pendent path P _k+1 to one vertex of C _n−k .