Bert Randerath

Vertex Colouring and Forbidden Subgraphs – A Survey

Abstract.There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions. Thus, one who wishes to obtain useful results from a graph coloring formulation of his problem must do more than just show that the problem is equivalent to the general problem of coloring a graph. If there is to be any hope, one must also obtain information about the structure of the graphs that need to be colored (D.S. Johnson [66]).

3-Colorability ∈ P for P 6 -free graphs

In this paper, we study a chromatic aspect for the class of P6-free graphs. Here, the focus of our interest are graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colorability can be decided in polynomial time and, if so, a proper 3-coloring can be determined also in polynomial time. Note that the 3-colorability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. for triangle- and K1,5-free graphs (Discrete Math. 162 (1-3) (1996) 313-317). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-coloring of a given graph in general. Our approach is based on an encoding of the problem with Boolean formulas making use of the existence of bounded dominating subgraphs. Together with a structural analysis of the nonperfect K4-free members of the graph class in consideration we obtain our main result that 3-colorability can be decided in polynomial time for the class of P6-free graphs.

3-Colorability ∈ P for P6-free graphs

In this paper, we study a chromatic aspect for the class of P6-free graphs. Here, the focus of our interest are graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colorability can be decided in polynomial time and, if so, a proper 3-coloring can be determined also in polynomial time. Note that the 3-colorability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. for triangle- and K1,5-free graphs (Discrete Math. 162 (1-3) (1996) 313-317). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-coloring of a given graph in general. Our approach is based on an encoding of the problem with Boolean formulas making use of the existence of bounded dominating subgraphs. Together with a structural analysis of the nonperfect K4-free members of the graph class in consideration we obtain our main result that 3-colorability can be decided in polynomial time for the class of P6-free graphs.

Three-colourbility and forbidden subgraphs. II: polynomial algorithms

In this paper we study the chromatic number for graphs with forbidden induced subgraphs. We focus our interest on graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colourability can be decided in polynomial time and, if so, a proper 3-colouring can be determined also in polynomial time. Note that the 3-colourability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. triangle-free and K1,5-free (Discrete Math. 162 (1-3) (1996) 313). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-colouring of a given graph in general. We present three different approaches to reach our goal. The first approach is purely a structural analysis of the graph class in consideration; the second one is a structural analysis of only the non-perfect K4-free members of the considered graph class; finally the last approach is based on propositional logic and bounded dominating subgraphs.

On the complexity of 4-coloring graphs without long induced paths

We show that deciding if a graph without induced paths on nine vertices can be colored with 4 colors is an NP-complete problem, improving a previous NP-completeness result proved by Woeginger and Sgall in 2001. The complexity of 4-coloring graphs without induced paths on five vertices remains open. We show that deciding if a graph without induced paths or cycles on five vertices can be colored with 4 colors can be done in polynomial time. A step in our algorithm uses the well-known and deep fact due to Grotschel, Lovasz and Schrijver stating that perfect graphs can be optimally colored in polynomial time.

Vertex pancyclic graphs

Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3 ≤ k ≤ n, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3 ≤ k ≤ n. In this paper, we shall present different sufficient conditions for graphs to be vertex pancyclic.

Characterization of graphs with equal domination and covering number

Abstract Let G be a simple graph of order n ( G ). A vertex set D of G is dominating if every vertex not in D is adjacent to some vertex in D , and D is a covering if every edge of G has at least one end in D . The domination number γ ( G ) is the minimum order of a dominating set, and the covering number β ( G ) is the minimum order of a covering set in G . In 1981, Laskar and Walikar raised the question of characterizing those connected graphs for which γ ( G ) = β ( G ). It is the purpose of this paper to give a complete solution of this problem. This solution shows that the recognition problem, whether a connected graph G has the property γ ( G ) = β ( G ), is solvable in polynomial time. As an application of our main results we determine all connected extremal graphs in the well-known inequality γ(G) ⩽ [ n(G) 2 ] of Ore (1962), which extends considerable a result of Payan and Xuong from 1982. With a completely different method, independently around the same time, Cockayne, Haynes and Hedetniemi also characterized the connected graphs G with γ(G) = [ n(G) 2 ] .

A Satisfiability Formulation of Problems on Level Graphs

Abstract Abstract In this note we present a formulation of two related combinatorial embedding problems concerning level graphs in terms of CNF-formulas. The first problem is known as level planar embedding and the second as crossing-minimization-problem.

3-Colorability and forbidden subgraphs. I: Characterizing pairs

In this paper we investigate chromatic aspects for graphs with forbidden induced subgraphs with emphasis on the question of 3-colorability. In the main part all possible pairs (A,B) of forbidden induced subgraphs, s.t. every A- and B-free graph is 3-colorable, are determined. Moreover, this is embedded in a more general problem and the main result extends Vizings classical edge coloring result.

On stable cutsets in line graphs

We answer a question of Brandstadt et al. by showing that deciding whether a line graph with maximum degree 5 has a stable cutset is NP-complete. Conversely, the existence of a stable cutset in a line graph with maximum degree at most 4 can be decided efficiently. The proof of our NP-completeness result is based on a refinement on a result due to Chvatal that recognizing decomposable graphs with maximum degree 4 is an NP-complete problem. Here, a graph is decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v. We also discuss some open problems on stable cutsets.