The set of problems we consider here are generalizations of square-free sequences [A. Thue, Über unendliche Zeichenreichen, Norske Vid Selsk. Skr. I. Mat. Nat. Kl. Christiana 7 (1906) 1–22]. A finite sequence a1a2 . . . an of symbols from a set S is called square-free if it does not contain a sequence of the form ww = x1x2 . . . xmx1x2 . . . xm, xi ∈ S, as a subsequence of consecutive terms. Ex...

Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use a structural parameters of the inp...

Given a graph G = (V,E) and positive integral vertex weights w : V → N, the max-coloring problem seeks to find a proper vertex coloring of G whose color classes C1, C2, . . . , Ck, minimize ∑k i=1 maxv∈Ciw(v). The problem arises in scheduling conflicting jobs in batches and in minimizing buffer size in dedicated memory managers. In this paper we present three approximation algorithms and one in...

In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and ∆ + 1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, compa...

An r l -system is an r-uniform hypergraph in which every set of l vertices lies in at most one edge. Let mk r l be the minimum number of edges in an r l -system that is not k-colorable. Using probabilistic techniques, we prove that ar l kr−1 ln k l/ l−1 ≤ mk r l ≤ br l kr−1 ln k l/ l−1 where br l is explicitly defined and ar l is sufficiently small. We also give a different argument proving (fo...

An equitable graph coloring is a proper vertex coloring, such that the size of the color classes differ by at most one. We present a flow-based scheme for generating pruning rules for enumerative algorithms such as DSATUR. The scheme models the extendability of a partial (equitable) coloring into an equitable coloring via network flows. Computational experiments show that significant reductions...

Total variant of well known graph coloring game is considered. We determine exact values of χ ′′ g for some classes of graphs and show that total game chromatic number is bounded from above by ∆ + 3∆ +. We also show relation between total game coloring number and game coloring index.

A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding ...

For a pair of integers 1Fgr, the g-chromatic number of an r-uniform Ž . hypergraph Hs V, E is the minimal k, for which there exists a partition of V into subsets < < T , . . . , T such that elT Fg for every egE. In this paper we determine the asymptotic 1 k i Ž . behavior of the g-chromatic number of the random r-uniform hypergraph H n, p for all r Ž yrq1. possible values of g and for all value...