Haitze J. Broersma

Toughness in graphs - A survey

In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. These categories are circumference, the disproof of the 2-tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness. We complete the survey with some tough open problems!

Updating the complexity status of coloring graphs without a fixed induced linear forest

A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph Pk denotes a path on k vertices. The @?-Coloring problem is the problem to decide whether a graph can be colored with at most @? colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P8-free graphs. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4-Coloring is NP-complete for P9-free graphs, and a result of Woeginger and Sgall, who showed that 5-Coloring is NP-complete for P8-free graphs. Additionally, we prove that the precoloring extension version of 4-Coloring is NP-complete for P7-free graphs, but that the precoloring extension version of 3-Coloring can be solved in polynomial time for (P2+P4)-free graphs, a subclass of P7-free graphs. Here P2+P4 denotes the disjoint union of a P2 and a P4. We denote the disjoint union of s copies of a P3 by sP3 and involve Ramsey numbers to prove that the precoloring extension version of 3-Coloring can be solved in polynomial time for sP3-free graphs for any fixed s. Combining our last two results with known results yields a complete complexity classification of (precoloring extension of) 3-Coloring for H-free graphs when H is a fixed graph on at most 6 vertices: the problem is polynomial-time solvable if H is a linear forest; otherwise it is NP-complete.

Three Complexity Results on Coloring Pk-Free Graphs

We prove three complexity results on vertex coloring problems restricted to P k -free graphs, i.e., graphs that do not contain a path on k vertices as an induced subgraph. First of all, we show that the pre-coloring extension version of 5-coloring remains NP-complete when restricted to P 6-free graphs. Recent results of Hoang et al. imply that this problem is polynomially solvable on P 5-free graphs. Secondly, we show that the pre-coloring extension version of 3-coloring is polynomially solvable for P 6-free graphs. This implies a simpler algorithm for checking the 3-colorability of P 6-free graphs than the algorithm given by Randerath and Schiermeyer. Finally, we prove that 6-coloring is NP-complete for P 7-free graphs. This problem was known to be polynomially solvable for P 5-free graphs and NP-complete for P 8-free graphs, so there remains one open case.

Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time

Let 2P3 denote the disjoint union of two paths on three vertices. A graph G that has no subgraph isomorphic to a graph H is called H-free. The Vertex Coloring problem is the problem to determine the chromatic number of a graph. Its computational complexity for triangle-free H-free graphs has been classified for every fixed graph H on at most 6 vertices except for the case H=2P3. This remaining case is posed as an open problem by Dabrowski, Lozin, Raman and Ries. We solve their open problem by showing polynomial-time solvability.

λ-backbone colorings along pairwise disjoint stars and matchings

Given an integer @l>=2, a graph G=(V,E) and a spanning subgraph H of G (the backbone of G), a @l-backbone coloring of (G,H) is a proper vertex coloring V->{1,2,...} of G, in which the colors assigned to adjacent vertices in H differ by at least @l. We study the case where the backbone is either a collection of pairwise disjoint stars or a matching. We show that for a star backbone S of G the minimum number @? for which a @l-backbone coloring of (G,S) with colors in {1,...,@?} exists can roughly differ by a multiplicative factor of at most [emailxa0protected] from the chromatic number @g(G). For the special case of matching backbones this factor is roughly [emailxa0protected]+1. We also show that the computational complexity of the problem Given a graph G with a star backbone S, and an integer @?, is there a @l-backbone coloring of (G,S) with colors in {1,...,@?}? jumps from polynomially solvable to NP-complete between @[emailxa0protected]+1 and @[emailxa0protected]+2 (the case @[emailxa0protected]+2 is even NP-complete for matchings). We finish the paper by discussing some open problems regarding planar graphs.

Complexity of conditional colorability of graphs

For positive integers

A general framework for coloring problems: old results, new results, and open problems

k

The computational complexity of the minimum weight processor assignment problem

and

Sharp Upper Bounds on the Minimum Number of Components of 2-factors in Claw-free Graphs

r

Path-kipas Ramsey numbers

, a conditional