On the Computational Complexity of the Forcing Chromatic Number (original) (raw)

A complexity dichotomy for critical values of the b-chromatic number of graphs

Theoretical Computer Science

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by χ b (G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on χ b (G): The maximum degree ∆(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i − 1. We obtain a dichotomy result stating that for fixed k ∈ {∆(G) + 1 − p, m(G) − p}, the problem is polynomial-time solvable whenever p ∈ {0, 1} and, even when k = 3, it is NP-complete whenever p ≥ 2. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree ∆(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by ∆(G). Second, we show that b-Coloring is FPT parameterized by ∆(G) + k (G), where k (G) denotes the number of vertices of degree at least k.

On the complexity of deciding whether the distinguishing chromatic number of a graph is at most two

Discrete Mathematics, 2011

In an article [3] published recently in this journal, it was shown that when k ≥ 3, the problem of deciding whether the distinguishing chromatic number of a graph is at most k is NP-hard. We consider the problem when k = 2. In regards to the issue of solvability in polynomial time, we show that the problem is at least as hard as graph automorphism but no harder than graph isomorphism.

The complexity of H-colouring of bounded degree graphs

Discrete Mathematics, 2000

We investigate the complexity of the H -colouring problem restricted to graphs of bounded degree. The H -colouring problem is a generalization of the standard c-colouring problem, whose restriction to bounded degree graphs remains NP-complete, as long as c is smaller than the degree bound (otherwise we can use the theorem of Brooks to obtain a polynomial time algorithm). For H -colouring of bounded degree graphs, while it is also the case that most problems are NP-complete, we point out that, surprisingly, there exist polynomial algorithms for several of these restricted colouring problems. Our main objective is to propose a conjecture about the complexity of certain cases of the problem. The conjecture states that for graphs of chromatic number three, all situations which are not solvable by the colouring algorithm inherent in the theorem of Brooks are NP-complete. We motivate the conjecture by proving several supporting results.

The Complexity of some Problems Related to Graph 3-colorability

Discrete Applied Mathematics, 1998

It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given graph has a stable set whose deletion results in (3) the complement of a bipartite graph, (4) a split graph or (5) a threshold graph. 0 1998 Elsevier Science B.V. All rights reserved.

On the complexity of the selective graph coloring problem in some special classes of graphs

Theoretical Computer Science, 2013

In this paper, we consider the selective graph coloring problem. Given an integer k ≥ 1 and a graph G = (V , E) with a partition V 1 , . . . , V p of V , it consists in deciding whether there exists a set V * in G such that |V * ∩ V i | = 1 for all i ∈ {1, . . . , p}, and such that the graph induced by V * is k-colorable. We investigate the complexity status of this problem in various classes of graphs.

Complexity of Coloring Graphs without Paths and Cycles

Lecture Notes in Computer Science, 2014

Let Pt and C denote a path on t vertices and a cycle on vertices, respectively. In this paper we study the k-coloring problem for (Pt, C)-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of P5-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that k-colorability of P5-free graphs for k ≥ 4 does not. These authors have also shown, aided by a computer search, that 4-colorability of (P5, C5)-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any k, the k-colorability of (P6, C4)-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for k = 3 and k = 4. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring (P6, C4)-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying; in fact they are not efficient in practice, as they depend on multiple use of Ramsey-type results and resulting tree decompositions of very high widths.) To complement these results we show that in most other cases the k-coloring problem for (Pt, C)free graphs is NP-complete. Specifically, for = 5 we show that k-coloring is NP-complete for (Pt, C5)-free graphs when k ≥ 4 and t ≥ 7; for ≥ 6 we show that k-coloring is NP-complete for (Pt, C)-free graphs when k ≥ 5, t ≥ 6; and additionally, for = 7, we show that k-coloring is also NP-complete for (Pt, C7)-free graphs if k = 4 and t ≥ 9. This is the first systematic study of the complexity of the k-coloring problem for (Pt, C)-free graphs. We almost completely classify the complexity for the cases when k ≥ 4, ≥ 4, and identify the last three open cases.

On the complexity of the circular chromatic number

Journal of Graph Theory, 2004

Circular chromatic number, χ c is a natural generalization of chromatic number. It is known that it is NP-hard to determine whether or not an arbitrary graph G satisfies χ(G) = χ c (G). In this paper we prove that this problem is NP-hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers k ≥ 2 and n ≥ 3, for a given graph G with χ(G) = n, it is NP-complete to verify if χ c (G) ≤ n − 1 k .

Improved hardness results for approximating the chromatic number

Proceedings of IEEE 36th Annual Foundations of Computer Science

We consider the minimum cut cover problem for a simple, undirected graphs G(V ; E): nd a minimum cardinality family of cuts C in G such that each edge e 2 E belongs to at least one cut C 2 C. The cardinality of the minimum cut cover of G is denoted by c(G). The motivation for this problem comes from testing of electronic component boards. Loulou has shown that the cardinality of a minimum cut cover in the complete graph is precisely dlog ne. However, determining the minimum cut cover of an arbitrary graph was posed as an open problem by Loulou. In this note we settle this open problem by showing that the cut cover problem is closely related to the graph coloring problem, thereby also obtaining a simple proof of Loulou's main result. We show that the problem is NP-complete in general, and moreover, the approximation version of this problem still remains NP-complete. Some other observations are made, all of which follow as a consequence of the close connection to graph coloring.

Algorithmic complexity of list colorings

Discrete Applied Mathematics, 1994

Given a graph G = (V, E) and a finite set L(u) at each vertex UE V, the List Coloring problem asks whether there exists a function f: