On color-critical ($P_{5},\overline{P}_5$)-free graphs (original) (raw)

Deciding k-colourability of P_5-free graphs in polynomial time

arXiv (Cornell University), 2007

The problem of computing the chromatic number of a P 5-free graph is known to be NP-hard. In contrast to this negative result, we show that determining whether or not a P 5-free graph admits a k-colouring, for each fixed number of colours k, can be done in polynomial time. If such a colouring exists, our algorithm produces it.

Deciding k-Colorability of P 5-Free Graphs in Polynomial Time

Algorithmica, 2010

A Refinement on the Structure of Vertex-Critical (P5, Gem)-Free Graphs

Social Science Research Network, 2022

We give a new, stronger proof that there are only finitely many k-vertex-critical (P 5 , gem)free graphs for all k. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all 6-and 7-vertexcritical (P 5 , gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the k-colourability of (P 5 , gem)-free graphs for all k where the certificate is either a k-colouring or a (k + 1)-vertex-critical induced subgraph. Our complete lists for k ≤ 7 allow for the implementation of these algorithms for all k ≤ 6.

Constructions ofk-criticalP5-free graphs

Discrete Applied Mathematics, 2015

With respect to a class C of graphs, a graph G ∈ C is said to be k-critical if every proper subgraph of G belonging to C is k−1 colorable. We construct an infinite set of k-critical P 5free graphs for every k ≥ 5. We also prove that there are exactly eight 5-critical {P 5 , C 5 }-free graphs.

On k-critical -free graphs

Electronic Notes in Discrete Mathematics, 2013

The Chromatic number problem is a classic problem in Computer Science, and is one of Karp's 21 NP-complete problems. The Chromatic number problem for P 5-free graphs is known to be NP-hard [7]. However for fixed k, the kcolorability question for P 5-free graphs can be answered in polynomial time [4,5]. We denote by P k (C k) the chordless path (cycle) on k vertices. More generally, the k-colorability question for P t-free graphs has been well studied [11,

The chromatic number of{P5,K4}-free graphs

Discrete Mathematics, 2013

Gyárfás conjectured that for any tree T every T-free graph G with maximum clique size ω(G) is f T (ω(G))-colorable, for some function f T that depends only on T and ω(G). Moreover, he proved the conjecture when T is the path P k on k vertices. In the case of P 5 , the best values or bounds known so far are f P 5 (2) = 3 and f P 5 (q) ≤ 3 q−1. We prove here that f P 5 (3) = 5.

4-colorability of P6-free graphs

Electronic Notes in Discrete Mathematics, 2015

In this paper we will study the complexity of 4-colorability in subclasses of P 6-free graphs. The well known k-colorability problem is NP-complete. It has been shown that if k-colorability is solvable in polynomial time for an induced H-free graph, then every component of H is a path. Recently, Huang [11] has shown several improved complexity results on k-coloring P t-free graphs, where P t is an induced path on t vertices. In summer 2014 only the case k = 4, t = 6 remained open for all k ≥ 4 and all t ≥ 6. Huang conjectures that 4-colorability of P 6-free graphs can be decided in polynomial time. This conjecture has shown to be true for the class of (P 6 , banner)-free graphs by Huang [11] and for the class of (P 6 , C 5)-free graphs by Chudnovsky et al. [6]. In this paper we show that the conjecture also holds for the class of (P 6 , bull, Z 2)-free graphs, for the class of (P 6 , bull, kite)-free graphs, and for the class of (P 6 , chair)-free graphs.

Dichotomizing k-vertex-critical H-free graphs for H of order four

Discrete Applied Mathematics, 2022

For k ≥ 3, we prove (i) there is a finite number of k-vertex-critical (P 2 + P 1)-free graphs and (ii) k-vertex-critical (P 3 + P 1)-free graphs have at most 2k − 1 vertices. Together with previous research, these results imply the following characterization where H is a graph of order four: There is a finite number of k-vertex-critical H-free graphs for fixed k ≥ 5 if and only if H is one of K 4 , P 4 , P 2 + 2P 1 , or P 3 + P 1. Our results imply the existence of new polynomial-time certifying algorithms for deciding the k-colorability of (P 2 + P 1)-free graphs for fixed k.

5-coloring K 3, k -minor-free graphs

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013

A seminal result of Thomassen [40] says that there are only finitely many 6-color-critical graphs for the bounded genus graphs. This result is no longer true if we consider K 3,k-minor-free graphs. K 3,k-minorfree graphs are a significant generalization of boundedgenus graphs. They also contain infinitely many t-colorcritical graphs for all t with k + 2 ≥ t ≥ 4. Motivated by this fact, we first show that if G is 6color-critical 4-connected K 3,k-minor-free graphs, then G has tree-width at most g(k) for some function g of k. This allows us to show the following algorithmic result, which is of independent interest. For a 4-connected graph G and any k, there is an O(n 3) algorithm to test whether or not G is 5-colorable. Note that testing the 3-colorability of K 3,kminor-free graphs is NP-complete, and testing the 4colorability of them would require a significant generalization of the Four Color Theorem (because K 3,3minor-free graphs are essentially planar). Testing the 5-colorability of bounded genus graphs can be done in polynomial time, as shown by Thomassen [40]. Thus our result can be viewed as a strengthening of the bounded genus case. We then investigate minimal forbidden minors in non-5-colorable K 3,k-minor-free graphs. Such a graph may exist, as K 6 shows. However, we prove the following result. There is a computable constant f (k) such that every minimal forbidden minor in non-5-colorable K 3,k-minorfree graphs has at most f (k) vertices. Our proof of the above result implies the following algorithmic result, which is of independent interest. For a graph G and any k, there is an O(n 3) algorithm to output one of the following:

On Indicated Coloring of Some Classes of Graphs

Graphs and Combinatorics, 2019

Indicated coloring is a type of game coloring in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by χ i (G). In this paper, we obtain structural characterization of connected {P 5 , K 4 , Kite, Bull}-free graphs which contains an induced C 5 and connected {P 6 , C 5 , K 1,3 }-free graphs that contains an induced C 6 . Also, we prove that {P 5 , K 4 , Kite, Bull}-free graphs that contains an induced C 5 and {P 6 , C 5 , P 5 , K 1,3 }free graphs which contains an induced C 6 are k-indicated colorable for all k ≥ χ(G). In addition, we show that K[C 5 ] is k-indicated colorable for all k ≥ χ(G) and as a consequence, we exhibit that {P 2 ∪ P 3 , C 4 }-free graphs, {P 5 , C 4 }-free graphs are k-indicated colorable for all k ≥ χ(G). This partially answers one of the questions which was raised by A. Grzesik in .

3-Colorability ∈P for P6-free graphs

Discrete Applied Mathematics, 2004

In this paper, we study a chromatic aspect for the class of P6-free graphs. Here, the focus of our interest are graph classes (deÿned 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 non-perfect 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.

Colorability of -free Graphs

2006

This paper considers the question of whether or not a -free graph can be 4-colored in polynomial time. It is known that a connected -free graph must have either a dominating clique or a dominating . Thus, when considering the 4-coloring question, we have three cases of interest: either has a dominating , a dominating , or a dominating . In this paper we demonstrate a polynomial time approach for determining whether or not a -free graph with a dominating can be 4-colored.

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.

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.

Chromatic Bounds for the Subclasses of pK2-Free Graphs

2021

In this paper, we study the chromatic number for graphs with forbidden induced subgraphs. We improve the existing χ-binding functions for some subclasses of 2K2-free graphs, namely {2K2,H}-free graphs where H ∈ {K5 − e,K2 + P4,K1 + C4}. In addition, for p ≥ 3, we find the polynomial χ-binding functions {pK2,H}-free graphs where H ∈ {gem, diamond,HV N,K5 − e,K2 + P4, butterf ly, dart, gem +, C4,K1 + C4, P5}.

Graph coloring satisfying restraints

Discrete Mathematics, 1990

For an integer k 2 2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k-colors. A graph G is amenably k-colorable if, for each nonconstant proper k-restraint r on G, there is a k-coloring c of G with c(v) # r(v) for each vertex v of G. A graph G is amenable if it is amenably k-colorable and k is the chromatic number of G. For any k Z= 3, there are infinitely many amenable k-critical graphs. For k 2 3, we use a construction of B. Toft and amenable graphs to associate a k-colorable graph to any k-colorable finite hypergraph. Some constructions for amenable graphs are given. We also consider a related property-being strongly critical-that is satisfied by many critical graphs, including complete graphs. A strongly critical graph is critical and amenable, but the converse is not always true. The Dirac join operation preserves both amenability and the strongly critical property. In addition, the Haj6s construction applied to a single edge in each of two strongly k-critical graphs yields an amenable graph. However, for any k 2 5, there are amenable k-critical graphs for which the Haj6s construction on two copies is not amenable.

New results on generalized graph coloring

2004

For graph classes P 1 , . . . , P k , Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V 1 , . . . ,V k so that V j induces a graph in the class P j ( j = 1, 2, . . . , k). If P 1 = · · · = P k is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k ≥ 3. Recently, this result has been generalized by showing that if all P i 's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs.