Elżbieta Sidorowicz

Dynamic Coloring of Graphs

Dynamics is an inherent feature of many real life systems so it is natural to define and investigate the properties of models that reflect their dynamic nature. Dynamic graph colorings can be naturally applied in system modeling, e.g. for scheduling threads of parallel programs, time sharing in wireless networks, session scheduling in high-speed LANs, channel assignment in WDM optical networks as well as traffic scheduling. In the dynamic setting of the problem, a graph we color is not given in advance and new vertices together with adjacent edges are revealed one after another at algorithms input during the coloring process. Moreover, independently of the algorithm, some vertices may lose their colors and the algorithm may be asked to color them again. We formally define a dynamic graph coloring problem, the dynamic chromatic number and prove various bounds on its value. We also analyze the effectiveness of the dynamic coloring algorithm Dynamic-Fit for selected classes of graphs. In particular, we deal with trees, products of graphs and classes of graphs for which Dynamic-Fit is competitive. Motivated by applications, we state the problem of dynamic coloring with discoloring constraints for which the performance of the dynamic algorithm Time-Fit is analyzed and give a characterization of graphs k-critical for Time-Fit. Since for any fixed k > 0 the number of such graphs is finite, it is possible to decide in polynomial time whether Time-Fit will always color a given graph with at most k colors.

Rainbow connection in oriented graphs

An edge-coloured graph G is said to be rainbow-connected if any two vertices are connected by a path whose edges have different colours. The rainbow connection number of a graph is the minimum number of colours needed to make the graph rainbow-connected. This graph parameter was introduced by G.?Chartrand, G.L.?Johns, K.A.?McKeon and P.?Zhang in 2008. Since, the topic drew much attention, and various similar parameters were introduced, all dealing with undirected graphs.Here, we initiate the study of rainbow connection in oriented graphs. An early statement is that the rainbow connection number of an oriented graph is lower bounded by its diameter and upper bounded by its order. We first characterize oriented graphs having rainbow connection number equal to their order. We then consider tournaments and prove that (i) the rainbow connection number of a tournament can take any value from 2 to its order minus one, and (ii) the rainbow connection number of every tournament with diameter d is at most d + 2 .

On the Non-(p−1)-Partite Kp-Free Graphs

Abstract We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(G) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p − 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m?

Weakly P-saturated graphs

For a hereditary property P let kP(G) denote the number of forbidden subgraphs contained in G. A graph G is said to be weakly Psaturated, if G has the property P and there is a sequence of edges of G, say e1, e2, . . . , el, such that the chain of graphs G = G0 ⊂ G0+e1 ⊂ G1 + e2 ⊂ . . . ⊂ Gl−1 + el = Gl = Kn (Gi+1 = Gi + ei+1) has the following property: kP(Gi+1) > kP(Gi), 0 ≤ i ≤ l − 1. In this paper we shall investigate some properties of weakly saturated graphs. We will find upper bound for the minimum number of edges of weakly Dk-saturated graphs of order n. We shall determine the number wsat(n,P) for some hereditary properties.

Acyclic colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.

Extremal bipartite graphs with a unique k-factor

Given integers p > k > 0, we consider the following problem of extremal graph theory: How many edges can a bipartite graph of order 2p have, if it contains a unique k-factor? We show that a labeling of the vertices in each part exists, such that at each vertex the indices of its neighbours in the factor are either all greater or all smaller than those of its neighbours in the graph without the factor. This enables us to prove that every bipartite graph with a unique k-factor and maximal size has exactly 2k vertices of degree k and 2k vertices of degree |V (G)| 2 . As our main result we show that for k ≥ 1, p ≡ t (mod k), 0 ≤ t < k, ∗The results were proved while the author was working at the Lehrstuhl C für Mathematik, RWTH-Aachen. 182 A. Hoffmann, E. Sidorowicz and L. Volkmann a bipartite graph G of order 2p with a unique k-factor meets 2|E(G)| ≤ p(p + k)− t(k− t). Furthermore, we present the structure of extremal graphs.

Some extremal problems of graphs with local constraints

Let P be a family of graphs. A graph G is said to satisfy a property P locally if G[N(υ)] ∈ P for every υ ∈ V(G). The class of graphs that satisfies the property P locally will be denoted by L(P) and we shall call such a class a local property.Let P be a hereditary property. A graph is said to be maximal with respect to a hereditary property P (shortly P-maximal) if it belongs to P and none of its proper supergraphs of the same order has the property P. A graph is P-extremal if it has the maximum number of edges among all P-maximal graphs of given order. This number is denoted by ex(n, P). If the number of edges of a P-maximal graph of order n is minimum, then the graph is called P-saturated and its number of edges is denoted by sat(n, P).In this paper, we shall describe the numbers ex(n,L(Ok)) and ex(n,L(Jk)) for k ≥ 1. Also, we give sat(n,L(Ok)) and sat(n,L(Jk)) for k = 1,2.

Generalized domination, independence and irredudance in graphs

The purpose of this paper is to present some basic properties of P-dominating, P-independent, and P-irredundant sets in graphs which generalize well-known properties of dominating, independent and irredundant sets, respectively.

Colouring game and generalized colouring game on graphs with cut-vertices

For k ≥ 2 we define a class of graphs Hk = {G : every block of G has at most k vertices}. The class Hk contains among other graphs forests, Husimi trees, line graphs of forests, cactus graphs. We consider the colouring game and the generalized colouring game on graphs from Hk.

Strong rainbow connection in digraphs

Abstract An arc-coloured digraph is strongly rainbow connected if for every pair of vertices ( u , v ) there exists a shortest path from u to v all of whose arcs have different colours. The strong rainbow connection number of a digraph is the minimum number of colours needed to make the graph strongly rainbow connected. In this paper, we study the strong rainbow connection number of minimally strongly connected digraphs, non-Hamiltonian strong digraphs and strong tournaments.