On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica 17 (1) (1997) 1-9] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n−O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasi-planar topological graph (i.e., every pair of edges have at most one point in common) is 6.5n−O(1), thereby exhibiting that non-simple quasi-planar graphs may have many more edges than simple ones.     

On edges crossing few other edges in simple topological complete graphs

We study the existence of edges having few crossings with the other edges in drawings of the complete graph (more precisely, in simple topological complete graphs). A topological graphT=(V,E) is a graph drawn in the plane with vertices represented by distinct points and edges represented by Jordan curves connecting the corresponding pairs of points (vertices), passing through no other vertices, and having the property that any intersection point of two edges is either a common end-point or a point where the two edges properly cross. A topological graph is simple if any two edges meet in at most one common point.Let h=h(n) be the smallest integer such that every simple topological complete graph on n vertices contains an edge crossing at most h other edges. We show that Ω(n^3/2)≤h(n)≤O(n²/log^1/4n). We also show that the analogous function on other surfaces (torus, Klein bottle) grows as cn².     

An optimal pram algorithm for a spanning tree on trapezoid graphs

LetG be a graph withn vertices andm edges. The problem of constructing a spanning tree is to find a connected subgraph ofG withn vertices andn?1 edges. In this paper, we propose anO(logn) time parallel algorithm withO(n/logn), processors on an EREW PRAM for constructing a spanning tree on trapezoid graphs.     

An implicit representation of chordal comparability graphs in linear time

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined in O(1) time whether they are adjacent, no matter how dense the graph is. We give a linear time algorithm for finding the four linear orders, improving on their bound of O(n²).     

On vertex symmetric digraphs

The problem of how “near” we can come to a n-coloring of a given graph is investigated. I.e., what is the minimum possible number of edges joining equicolored vertices if we color the vertices of a given graph with n colors. In its generality the problem of finding such an optimal color assignment to the vertices (given the graph and the number of colors) is NP-complete. For each graph G, however, colors can be assigned to the vertices in such a way that the number of offending edges is less than the total number of edges divided by the number of colors. Furthermore, an Ω(epn) deterministic algorithm for finding such an n-color assignment is exhibited where e is the number of edges and p is the number of vertices of the graph (e?p?n). A priori solutions for the minimal number of offending edges are given for complete graphs; similarly for equicolored K_m in K_p and equicolored graphs in K_p.     

The backup 2-median problem on block graphs

The backup 2-median problem is a location problem to locate two facilities at vertices with the minimum expected cost where each facility may fail with a given probability. Once a facility fails, the other one takes full responsibility for the services. Here we assume that the facilities do not fail simultaneously. In this paper, we consider the backup 2-median problem on block graphs where any two edges in one block have the same length and the lengths of edges on different blocks may be different. By constructing a tree-shaped skeleton of a block graph, we devise an O（n log n q- m）-time algorithm to solve this problem where n and m are the number of vertices and edges, respectively, in the given block graph.     

A Classification of the Strongly Regular Generalised Johnson Graphs

The generalised Johnson graphs are the graphs J(n, k, m) whose vertices are the k subsets of {1, 2, . . . , n}, with two vertices J ₁ and J ₂ joined by an edge if and only if ${{|J_1 \cap J_2| = m}}$ . A graph is called d-regular if every vertex has exactly d edges incident to it. A d-regular graph on v vertices is called a (v, d, a, c)-strongly regular graph if every pair of adjacent vertices have exactly a common neighbours and every pair of non-adjacent vertices have exactly c common neighbours. The triangular graphs J(n, 2, 1), their complements J(n, 2, 0), the sporadic examples J(10, 3, 1) and J(7, 3, 1), as well as the trivially strongly regular graphs J(2k, k, 0) are examples of strongly regular generalised Johnson graphs. In this paper we prove that there are no other strongly regular generalised Johnson graphs.     

Intersection reverse sequences and geometric applications

Pinchasi and Radoi?i? [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C₄ has O(n^3/2logn) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n^3/2logn) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.     

An optimal algorithm for finding depth-first spanning tree on permutation graphs

LetG be a connected graph ofn vertices. The problem of finding a depth-first spanning tree ofG is to find a connected subgraph ofG with then vertices andn − 1 edges by depth-first-search. In this paper, we propose anO(n) time algorithm to solve this problem on permutation graphs.     

Binary tree algebraic computation and parallel algorithms for simple graphs

In this paper we define the binary tree algebraic computation (BTAC) problem and develop an efficient parallel algorithm for solving this problem. A variety of graph problems (minimum covering set, minimum r-dominating set, maximum matching set, etc.) for trees and two terminal series parallel (TTSP) graphs can be converted to instances of the BTAC problem. Thus efficient parallel algorithms for these problems are obtained systematically by using the BTAC algorithm. The parallel computation model is an exclusive read exclusive write PRAM. The algorithms for tree problems run in O(log n) time with O(n) processors. The algorithms for TTSP graph problems run in O(log m) time with O(m) processors where n (m) is the number of vertices (edges) in the input graph. These algorithms are within an O(log n) factor of optimal.     

A trust region method for solving the decentralized static output feedback design problem

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n ²) time sequential algorithm and anO(n ²/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.     

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) ≤  r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (K _1,n for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) ≤  rk. Hitherto, the only reported upper bound on the rainbow connection number of bridgeless graphs is 4n/5 ? 1, where n is order of the graph (Caro et al. in Electron J Comb 15(1):Research paper 57, 13, 2008). It is known that computing rc(G) is NP-Hard (Chakraborty and fischer in J Comb Optim 1–18, 2009). Here, we present a (r + 3)-factor approximation algorithm which runs in O(nm) time and a (d + 3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.     

On bipartite graphs with minimal energy

The energy of a graph is the sum of the absolute values of the eigenvalues of the graph. In a paper [G. Caporossi, D. Cvetkovi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996] Caporossi et al. conjectured that among all connected graphs G with n≥6 vertices and n−1≤m≤2(n−2) edges, the graphs with minimum energy are the star S_n with m−n+1 additional edges all connected to the same vertices for m≤n+⌊(n−7)/2⌋, and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. The conjecture is proved to be true for m=n−1,2(n−2) in the same paper by Caporossi et al. themselves, and for m=n by Hou in [Y. Hou, Unicyclic graphs with minimal energy, J. Math. Chem. 29 (2001) 163-168]. In this paper, we give a complete solution for the second part of the conjecture on bipartite graphs. Moreover, we determine the graph with the second-minimal energy in all connected bipartite graphs with n vertices and edges.     

Generating 3-vertex connected spanning subgraphs

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K²log(K)m²+K²m³) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.     

On a conjecture of Brouwer involving the connectivity of strongly regular graphs

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components.We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called Δ-spaces are counterexamples to Brouwer?s Conjecture. Using J.I. Hall?s characterization of finite reduced copolar spaces, we find that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field Fq (for any q prime power), and the strongly regular graphs constructed from the hyperbolic quadrics O⁺(2r,2) and from the elliptic quadrics O⁻(2r,2) over the field F₂, respectively, are counterexamples to Brouwer?s Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall?s characterization theorem for Δ-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of Δ-spaces and thus, yield other counterexamples to Brouwer?s Conjecture.We prove that Brouwer?s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles GQ(q,q) graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue −2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components.     

On the Laplacian spectral radii of bicyclic graphs

A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n+1. Let B(n) be the set of all bicyclic graphs on n vertices. In this paper, we obtain the first four largest Laplacian spectral radii among all the graphs in the class B(n) (n≥7) together with the corresponding graphs.     

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x∈V(G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(mlogn) time whether G is a median graph with geodetic number 2.     

The extremal graph problem of the icosahedron

P. Turán has asked the following question:Let I¹² be the graph determined by the vertices and edges of an icosahedron. What is the maximum number of edges of a graph Gⁿ of n vertices if Gⁿ does not contain I¹² as a subgraph?We shall answer this question by proving that if n is sufficiently large, then there exists only one graph having maximum number of edges among the graphs of n vertices and not containing I¹². This graph Hⁿ can be defined in the following way:Let us divide n ? 2 vertices into 3 classes each of which contains [$\text{(n?2)}\text{3}$] or $\text{[}\text{(n?2)}\text{3}\text{] + 1}$ vertices. Join two vertices iff they are in different classes. Join two vertices outside of these classes to each other and to every vertex of these three classes.     

Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. In particular, we prove that if C(n) denotes the largest number of touching pairs in a packing of n>1 congruent balls in Euclidean 3-space, then $0.695<\frac{6n-C(n)}{n^{\frac{2}{3}}}< \sqrt[3]{486}=7.862\dots$ for all $n=\frac{k(2k^{2}+1)}{3}$ with k??2.     

On the Number of Edges in Colour-Critical Graphs and Hypergraphs

A (hyper)graph G is called k-critical if it has chromatic number k, but every proper sub(hyper)graph of it is (k-1)-colourable. We prove that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k-o(k))n edges. Furthermore, we show that every (k+1)-critical hypergraph on n vertices and without graph edges has at least (k-3/³?{k}) n(k-3/\sqrt[3]{k}) n edges. Both bounds differ from the best possible bounds by o(kn) even for graphs or hypergraphs of arbitrary girth.