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.     

Noncrossing Hamiltonian paths in geometric graphs

A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: determine the largest number h(n) such that when we remove any set of h(n) edges from any complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that . We also establish several results related to special classes of geometric graphs. Let h₁(n) denote the largest number such that when we remove edges of an arbitrary complete subgraph of size at most h₁(n) from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We prove that . Let h₂(n) denote the largest number such that when we remove an arbitrary star with at most h₂(n) edges from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We show that h₂(n)=⌈n/2⌉-1. Further we prove that when we remove any matching from a complete geometric graph the resulting graph will have a noncrossing Hamiltonian path.     

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.     

A note on similar edges and edge-unique line graphs

If G is a connected graph having no vertices of degree 2 and L(G) is its line graph, two results are proven: if there exist distinct edges e and f with L(G) ? e ? L(G) ? f then there is an automorphism of L(G) mapping e to f; if $\text{G ? u ¦ G ? v}$ for any distinct vertices u, v, then $\text{L(G) ? e ¦ L(G) ? f}$ for any distinct edges e, f.     

Enumeration of simple complete topological graphs

A simple topological graph T=(V(T),E(T)) is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is 2^Θ(n4). We also show that the number of weak isomorphism classes of simple complete topological graphs with n vertices and crossings is at least 2^{n(logn−O(1))}, which improves the estimate of Harborth and Mengersen.     

Geometric Graphs with Few Disjoint Edges

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and T?rőcsik, we show that a geometric graph on n vertices with no k+1 pairwise disjoint edges has at most k ³ (n+1) edges. On the other hand, we construct geometric graphs with n vertices and approximately (3/2)(k-1)n edges, containing no k+1 pairwise disjoint edges. We also improve both the lower and upper bounds of Goddard, Katchalski, and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges. Received May 7, 1998, and in revised form March 24, 1999.     

Bounds on the eigenvalues of graphs with cut vertices or edges

In this paper, we characterize the extremal graph having the maximal Laplacian spectral radius among the connected bipartite graphs with n vertices and k cut vertices, and describe the extremal graph having the minimal least eigenvalue of the adjacency matrices of all the connected graphs with n vertices and k cut edges. We also present lower bounds on the least eigenvalue in terms of the number of cut vertices or cut edges and upper bounds on the Laplacian spectral radius in terms of the number of cut vertices.     

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     

On the maximum number of edges in a hypergraph whose linegraph contains no cycle

Soit H = (X,F) un hypergraphe h-uniforme avec ∥X∥ = n et soit L_h±1(H) le graphe dont les sommets représentent les arêtes de H. deux sommets étant relíes si et seulement si les arétes qu'ils représentent intersectent en h ± 1 sommets. Nous montrons que si L_h±1(H) ne contient pas de cycle, alors $\text{∥F∥<(}{}^{\mathrm{n}}{}_{\mathrm{h\pm 1}}\text{)/h±1}$. la borne étant exacte pour h = 2 et pour des valeurs de H pour h = 3. Ce probl`eme mène á une conjecture sur les “presque systèmes de Steine.”Let H = (X, F) be a h-uniform hypergraph, with ∥X∥ = n and let L_h±1(H) be the graph whose vertices are the edges of H, two vertices being joined if and only if the edges they represent intersect in h ±1 vertices. We prove that, if L_h±1H contains no cycle, then $\text{∥F∥(}{}^{\mathrm{n}}{}_{\mathrm{h\pm 1}}\text{)/h±1}$; moreover the bound is exact for h = 2 and with some values of n for h = 3. This problem leads to a conjecture on “almost Steiner systems”.     

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.     

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.     

On the Structure of Graphs with Low Obstacle Number

The obstacle number of a graph G is the smallest number of polygonal obstacles in the plane with the property that the vertices of G can be represented by distinct points such that two of them see each other if and only if the corresponding vertices are joined by an edge. We list three small graphs that require more than one obstacle. Using extremal graph theoretic tools developed by Pr?mel, Steger, Bollobás, Thomason, and others, we deduce that for any fixed integer h, the total number of graphs on n vertices with obstacle number at most h is at most 2^o(n2){2^{o(n^2)}}. This implies that there are bipartite graphs with arbitrarily large obstacle number, which answers a question of Alpert et al. (Discret Comput Geom doi:, 2009).     

Cycle and cocycle coverings of graphs

In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family ?? of at most n?1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and ε edges having cogirth g^*?3 and k(G) components, there is a family of at most ε?n+k(G) cocycles which cover the edges of G at least twice. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 270–284, 2010     

A Turán type problem for interval graphs

We consider the following analogue of a problem of Turán for interval graphs: Let c = c(n, m) be the largest integer such that any interval graph with n vertices and at least m edges contains a complete subgraph on c vertices. We determine the value of c(n, m) explicitly.     

Duality and perfection for edges in cliques

Given a graph G, let K(G) denote the graph whose vertices correspond with the edges of G. Two vertices of K(G) are joined by an edge if the corresponding edges in G are contained in a clique. This paper investigates some properties of G which force duality theorems for K(G).     

On a problem of C. C. Chen and D. E. Daykin

Let α(k, p, h) be the maximum number of vertices a complete edge-colored graph may have with no color appearing more than k times at any vertex and not containing a complete subgraph on p vertices with no color appearing more than h times at any vertex. We prove that $\text{α(k, p, h) ≤ h + 1 + (k ? 1){(p ? h ? 1) × (h}{}^{\mathrm{p}}\text{+ 1)}}{}^{\mathrm{<mtext>1</mtext><mtext>h</mtext>}}$ and obtain a stronger upper bound for α(k, 3, 1). Further, we prove that a complete edge-colored graph with n vertices contains a complete subgraph on p vertices in which no two edges have the same color if

$\text{(}\text{n}\text{3}\text{)>(}\text{p}\text{3}\text{)}\text{Σ}\text{i=1}\text{t}\text{(}\text{e}{}_{\mathrm{i}}\text{2}\text{)}$

where e_i is the number of edges of color i, 1 ≤ i ≤ t.     

Large dense neighbourhoods and turán's theorem

Let t_r(n) denote the number of edges in the complete r-partite graph on n vertices whose colour classes are as equal in size as possible. It is proved that if G is a simple graph on n vertices and more than t_r(n) edges, and if v is any vertex of maximum degree m in G, then the subgraph of G induced by the neighbours of v has more than t_r?1(m) edges. Examples are given to demonstrate the sharpness of this theorem, and its algorithmic implications are noted.     

Embedding Planar Graphs at Fixed Vertex Locations

 Let G be a planar graph of n vertices, v ₁,…,v _n , and let {p ₁,…,p _n } be a set of n points in the plane. We present an algorithm for constructing in O(n ²) time a planar embedding of G, where vertex v _i is represented by point p _i and each edge is represented by a polygonal curve with O(n) bends (internal vertices). This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of G mapping vertices to their assigned points and edges to polygonal curves has at least m/20 edges represented by curves with at least m/40³ bends. Received: May 24, 1999 Final version received: April 10, 2000     

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt _p-1( ⁿ ) cliques. Heret _p-1( ⁿ ) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp. In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn ²/2. We prove this conjecture forK ₄-free graphs and show that in the case of equality forC andG there are only two possibilities:

1. G?K _n/2,n/2
2. G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.     

2-Approximation algorithm for finding a clique with minimum weight of vertices and edges

The problem of finding a minimum clique (with respect to the total weight of its vertices and edges) of fixed size in a complete undirected weighted graph is considered along with some of its important subclasses. Approximability issues are analyzed. The inapproximability of the problem is proved for the general case. A 2-approximation efficient algorithm with time complexity O(n ²) is suggested for the cases when vertex weights are nonnegative and edge weights either satisfy the triangle inequality or are squared pairwise distances for some point configuration of Euclidean space.