Michał Małafiejski

Consecutive colorings of the edges of general graphs

Abstract Given an n -vertex graph G , an edge-coloring of G with natural numbers is a consecutive (or interval) coloring if the colors of edges incident with each vertex are distinct and form an interval of integers. In this paper we prove that if G has a consecutive coloring and n⩾3 then S(G)⩽2n−4 , where S(G) is the maximum number of colors allowing a consecutive coloring. Next, we investigate the so-called deficiency of G , a natural measure of how far it falls of being consecutively colorable. Informally, we define the deficiency def (G) of G as the minimum number of pendant edges which would need to be attached in order that the resulting supergraph has such a coloring, and compute this number in the case of cycles, wheels and complete graphs.

On the deficiency of bipartite graphs

Abstract Given a graph G, an edge-coloring of G with colors 1,2,3,… is consecutive if the colors of edges incident to each vertex form an interval of integers. This paper is devoted to bipartite graphs which do not have such a coloring of edges. We investigate their consecutive coloring deficiency, or shortly the deficiency d(G) of G, i.e. the minimum number of pendant edges whose attachment to G makes it consecutively colorable. In particular, we show that there are bipartite graphs whose deficiency approaches the number of vertices.

Weakly cooperative guards in grids

We show that a minimum coverage of a grid of n segments has n–p3 weakly cooperative guards, where p3 is the size of the maximum P3-matching in the intersection graph of the grid. This makes the minimum weakly cooperative guards problem in grids NP-hard, as we prove that the maximum P3-matching problem in subcubic bipartite planar graphs is NP-hard. At last, we propose a 7/6-approximation algorithm for the minimum weakly cooperative guards problem.

A 27/26-Approximation Algorithm for the Chromatic Sum Coloring of Bipartite Graphs

We consider the CHROMATIC SUM PROBLEM on bipartite graphs which appears to be much harder than the classical CHROMATIC NUMBER PROBLEM. We prove that the CHROMATIC SUM PROBLEM is NP-complete on planar bipartite graphs with ? ? 5, but polynomial on bipartite graphs with ? ? 3, for which we construct an O(n2)-time algorithm. Hence, we tighten the borderline of intractability for this problem on bipartite graphs with bounded degree, namely: the case ? = 3 is easy, ? = 5 is hard. Moreover, we construct a 27/26-approximation algorithm for this problem thus improving the best known approximation ratio of 10/9.

The complexity of the L(p,q) -labeling problem for bipartite planar graphs of small degree

Given a simple graph G, by an L(p,q)-labeling of G we mean a function c that assigns nonnegative integers to its vertices in such a way that if two vertices u, v are adjacent then |c(u)-c(v)|>=p, and if they are at distance 2 then |c(u)-c(v)|>=q. The L(p,q)-labeling problem can be defined as follows: given a graph G and integer t, determine whether there exists an L(p,q)-labeling c of G such that c(V)@?{0,1,...,t}. In the paper we show that the problem is NP-complete even when restricted to bipartite planar graphs of small maximum degree and for relatively small values of t. More precisely, we prove that: (1)if p 3q then the problem is NP-complete for bipartite planar graphs of maximum degree @D@?4 and t=p+5q. In particular, these results imply that the L(2,1)-labeling problem in planar graphs is NP-complete for t=4, and that the L(p,q)-labeling problem in graphs of maximum degree @D@?4 is NP-complete for all values of p and q, thus answering two well-known open questions.

Application of an online judge & contester system in academic tuition

The paper contains a description of the SPOJ online judge and contester system, used for E-Learning of programming, which has been successfully applied in the tuition of students at the Gdansk University of Technology. We study the implementation of the system with security demands and present our experiences connected with the use of such systems in academic courses at an undergraduate and graduate level in the last four years.

Cooperative mobile guards in grids

A grid P is a connected union of vertical and horizontal segments. A mobile guard is a guard which is allowed to move along a grid segment, thus a point x is seen by a mobile guard g if either x is on the same segment as g or x is on a grid segment crossing g. A set of mobile guards is weakly cooperative if at any point on its patrol, every guard can be seen by at least one other guard. In this paper we discuss the classes of polygon-bounded grids and simple grids for which we propose a quadratic time algorithm for solving the problem of finding the minimum weakly cooperative guard set (MinWCMG). We also provide an O(nlogn) time algorithm for the MinWCMG problem in horizontally or vertically unobstructed grids. Next, we investigate complete rectangular grids with obstacles. We show that as long as both dimensions of a grid are larger than the number of obstacles k, k+2 weakly cooperative mobile guards always suffice to cover the grid. Finally, we prove that the MinWCMG problem is NP-hard even for grids in which every segment crosses at most three other segments. Consequently, the minimum k-periscope guard problem for 2D grids is NP-hard as well, and this answers the question posed by Gewali and Ntafos [L.P. Gewali, S. Ntafos, Covering grids and orthogonal polygons with periscope guards, Computational Geometry: Theory and Applications 2 (1993) 309-334].

Packing [1, Δ]-factors in graphs of small degree

Given an undirected, connected graph G with maximum degree Δ, we introduce the concept of a [1, Δ]-factor k-packing in G, defined as a set of k edge-disjoint subgraphs of G such that every vertex of G has an incident edge in at least one subgraph. The problem of deciding whether a graph admits a [1,Δ]-factor k-packing is shown to be solvable in linear time for k = 2, but NP-complete for all k≥ 3. For k = 2, the optimisation problem of minimising the total number of edges of the subgraphs of the packing is NP-hard even when restricted to subcubic planar graphs, but can in general be approximated within a factor of

An approximation algorithm for maximum P 3 -packing in subcubic graphs

\frac{42\Delta -30}{35\Delta-21}