Mihály Hujter

Precoloring extension. I: Interval graphs

Abstract This paper is the first article in a series devoted to the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this ‘precoloring’ be extended to a proper coloring of G with at most k colors (for some given k )? This question was motivated by practical problems in scheduling and VLSI theory. Here we investigate its complexity status for interval graphs and for graphs with a bounded treewidth.

Precoloring Extension 3: Classes of Perfect Graphs.

We continue the study of the following general problem on the vertex colourings of graphs. Suppose that some vertices of a graph G are assigned to some colours. Can this ‘precolouring’ be extended to a proper colouring of G with at most k colours (for some given k )? Here we investigate the complexity status of precolouring extendibility on some classes of perfect graphs, giving good characterizations (necessary and sufficient conditions) that lead to algorithms with linear or polynomial running time. It is also shown how a larger subclass of perfect graphs can be derived from graphs containing no induced path on four vertices.

Graphs with no induced C 4 and 2 K 2

Abstract We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class G of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members of G , the number of cliques (as well as the number of maximal independent sets) cannot exceed the number of vertices. Moreover, these graphs are almost extremal to the theorem of Nordhaus and Gaddum (1956).

An upper bound on the number of cliques in a graph

Giving a partial solution to a conjecture of Balas and Yu [Networks19 (1989) 247–235], we prove that if the complement of a graph G on n vertices contains no set of t + 1 pairwise disjoint edges as an induced subgraph, then G has fewer than (n/2t)2t maximal complete subgraphs.

A note on the complexity of the transportation problem with a permutable demand vector

Abstract. In this note we investigate the computational complexity of the transportation problem with a permutable demand vector, TP-PD for short. In the TP-PD, the goal is to permute the elements of the given integer demand vector b=(b1,…,bn) in order to minimize the overall transportation costs. Meusel and Burkard [6] recently proved that the TP-PD is strongly NP-hard. In their NP-hardness reduction, the used demand values bj, j=1,…,n, are large integers. In this note we show that the TP-PD remains strongly NP-hard even for the case where bj∈{0,3} for j=1,…,n. As a positive result, we show that the TP-PD becomes strongly polynomial time solvable if bj∈{0,1,2} holds for j=1,…,n. This result can be extended to the case where bj∈{κ,κ+1,κ+2} for an integer κ.

On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d > 0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.

Some numerical problems in discrete geometry

Abstract What is the smallest square which contains ten pairwise disjoint congruent open disks of unit diameter? It is conjectured that the minimum side is 3.373.... This paper proves that the side is at least 3.334.

Improving a method of search for solving polynomial equations

Abstract This paper is related to the Lehmer-Schur methods in numerical mathematics in the complex plane. It is shown that by a slight modification of the “optimized” Lehmer-Schur method of Galantai, the “speed” quotient 0.6094 can be reduced to 0.5758. The crucial idea is based on a discrete geometrical observation

Minimum order of graphs with given coloring parameters

A complete k -coloring of a graph G = ( V , E ) is an assignment ? : V ? { 1 , ? , k } of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number ? ( G ) and achromatic number ? ( G ) , respectively), and the Grundy number ? ( G ) defined as the largest k admitting a complete coloring ? with exactly k colors such that every vertex v ? V of color ? ( v ) has a neighbor of color i for all 1 ? i < ? ( v ) . The inequality chain ? ( G ) ? ? ( G ) ? ? ( G ) obviously holds for all graphs G . A triple ( f , g , h ) of positive integers at least 2 is called realizable if there exists a connected graph G with ? ( G ) = f , ? ( G ) = g , and ? ( G ) = h . In Chartrand et?al. (2010), the list of realizable triples has been found. In this paper we determine the minimum number of vertices in a connected graph with chromatic number f , Grundy number g , and achromatic number h , for all realizable triples ( f , g , h ) of integers. Furthermore, for f = g = 3 we describe the (two) extremal graphs for each h ? 6 . For h ? { 4 , 5 } , there are more extremal graphs, their description is given as well.

Some good characterization results relating to the Kőnig-Egerváry theorem

We survey some combinatorial results which are all related to some former results of ours, and, at the same time, they are all related to the famous Kőnig–Egerváry theorem from 1931.