Ervin Györi

The number of small semispaces of a finite set of points in the plane

Abstract For a configuration S of n points in the plane, let gk(S) denote the number of subsets of cardinality ⩽k cut off by a line. Let gk,n = max{gk(S): |S| = n}. Goodman and Pollack (J. Combin. Theory Ser. A 36 (1984) , 101–104) showed that if k n 2 then gk,n ⩽ 2nk − 2k2 − k. Here we show that gk,n = k·n for k n 2 .

Vertex-Disjoint Cycles Containing Specified Edges*

Abstract. Dirac and Ore-type degree conditions are given for a graph to contain vertex disjoint cycles each of which contains a previously specified edge. One set of conditions is given that imply vertex disjoint cycles of length at most 4, and another set of conditions are given that imply the existence of cycles that span all of the vertices of the graph (i.e. a 2-factor). The conditions are shown to be sharp and give positive answers to conjectures of Enomoto in [3] and Wang in [5].

C 6 -free bipartite graphs and product representation of squares

for any F > 0 if n is large enough (depending on k and E). The case k = 2 is nearly trivial since Fz(n) is equal to the number of square-free integers not exceeding n which is about (6/7c2)n. For the case of even integers k > 2, they worked out a number-theoretic method which reduces the problem to the study of Ck-free unbalanced bipartite graphs. For k = 4, the corresponding graph-theoretic problem can be settled relatively easily so that they got the following quite satisfactory estimates:

A minimax theorem on intervals

Abstract A problem of V. Chvatal is solved. It is proved that if P is a vertically convex lattice polygon with vertical and horizontal sides then the minimum number of the rectangles in P which cover P is equal to the maximum cardinality of the point sets such that any two elements induce a rectangle (with vertical and horizontal sides) not contained by the polygon P . This result is the best possible in some sense. Actually the following conjecture of A. Frank is proved which implies this theorem. A sequence of intervals I 1 ,…, I m is U -increasing if − u j=1 k−1 I j ≠ ∪ j=1 k I j for k = 2, 3,…, m . The set of intervals G is a generating set for the set of intervals J if every element of J is a union of members of G . We prove the conjecture that for any set of intervals J , the minimum size of a generating set G for J is equal to the maximum size of a U -increasing sequence of intervals with members from J .

Longest Paths in Circular Arc Graphs

We show that all maximum length paths in a connected circular arc graph have non-empty intersection.

Generalized guarding and partitioning for rectilinear polygons

Abstract A Tk-guard G in a rectilinear polygon P is a tree of diameter k completely contained in P. The guard G is said to cover a point x if x is visible from some point contained in G. We investigate the function r(n,h,k), which is the largest number of Tk-guards necessary to cover any rectilinear polygon with h holes and n vertices. The aim of this paper is to prove new lower and upper bounds on parts of this function. In particular, we show the following upper bounds: 1. r(n,0,k)⩽ n k+4  , with equality for even k. 2. r(n,h,1)= n+ 4h 3 + 4 3 4+ 4 3  3. (n,h,2)⩾ n 6 These bounds, along with other lower bounds that we establish, suggest that the presence of holes reduces the number of guards required, if k > 1. In the course of proving the upper bounds, new results on partitioning are obtained which also have efficient algorithmic versions.

Average distance in graphs with removed elements

The average distance in a graph is defined as the average length of a shortest path between two vertices, taken over all pairs of vertices. This parameter can be contrasted with the diameter as a description of the metrical properties of the graph. If a vertex or edge is removed from a graph, in the worst case the average distance will increase. In this paper we consider the question, How small can the increase be if we choose the vertex or edge appropriately? If the graph is a cycle, the increase is roughly by a factor of 4/3. P. Winkler conjectured that any graph contains a vertex whose removal will not increase the average distance by more than a factor of 4/3. A similar conjecture can be made about removed edges, provided, for example, that the graph is 2-edge-connected. We prove the stronger statement that the edge version of the conjecture is true if the graph has no vertices of degree one. We also give an asymptotic affirmative answer for the vertex version.

The Cartesian product of a k -extendable and an l -extendable graph is ( k + l +1)-extendable

Abstract : Let us start with the definition of a kappa-extendable graph G. Suppose kappa is an integer such that 1 < or = kappa < or = (/V(G)/-2)/2. A graph G is kappa-extendable if G is connected, has a perfect matching (a 1- factor) and any matching in G consisting of kappa edges can be extended to (i.e. , is a subset of) a perfect matching. The extendability number of G, extG, is the maximum kappa such that G is kappa-extendable. A natural problem is to determine the extendability number of a graph G.

Graphs without short odd cycles are nearly bipartite

Abstract It is proved that for every constant ϵ > 0 and every graph G on n vertices which contains no odd cycles of length smaller than ϵn , G can be made bipartite by removing (15/ϵ)ln(10/ϵ)) vertices. This result is best possible except for a constant factor. Moreover, it is shown that one candestroy all odd cycles in such a graph G also by omitting not more than (200/ ϵ 2 )(ln(10/ ϵ )) 2 edges.

On the maximal number of certain subgraphs inKr-free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) ≤ H(T2(n)), whereT2(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n ≥ max(m, r − 1), then there exists a complete (r − 1)-partite graphG* withn vertices such thatK(G) ≤ K(G*) holds for everyKr-free graphG withn vertices. In particular, in the class of allKr-free graphs withn vertices the complete balanced (r − 1)-partite graphTr−1(n) has the largest number of subgraphs isomorphic toKt (t < r),C4,K2,3. These generalize some theorems of Turán, Erdös and Sauer.