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Some problems on induced subgraphs
Abstract
We discuss some problems related to induced subgraphs. The first problem is about getting a good upper bound for the chromatic number in terms of the clique number for graphs in which every induced cycle has length 3 or 4. The second problem is about the perfect chromatic number of a graph, which is the smallest number of perfect sets into which the vertex set of a graph can be partitioned. (A set of vertices is said to be perfect it it induces a perfect graph.) The third problem is on antichains in the induced subgraph ordering. The fourth problem is on graphs in which the difference between the chromatic number and the clique number is at most one for every induced subgraph of the graph. The fifth problem is on a weakening of the notorious Erd\H{o}s-Hajnal conjecture. The last problem is on a conjecture of Gy\'{a}rf\'{a}s about -boundedness of a particular class of graphs.
... Hoáng and McDiarmid [8] conjectured for an odd hole free graph G, χ(G) ≤ 2 ω(G)−1 . A graph is said to be short-holed if every hole of it has length 4. Sivaraman [11] conjectured that χ(G) ≤ ω 2 (G) for all short-holed graphs whereas the best known upper-bound is χ(G) ≤ 10 20 2 ω 2 (G) due to Scott and Seymour [13]. ...
For a number , let denote the family of graphs which have girth 2l+1 and have no odd hole with length greater than 2l+1. Wu, Xu and Xu conjectured that every graph in is 3-colorable. Chudnovsky et al., Wu et al., and Chen showed that every graph in , and is 3-colorable respectively. In this paper, we prove that every graph in is 3-colorable. This confirms Wu, Xu and Xu's conjecture.
... Here are some open questions, different from those discussed earlier. (There are yet more in [100].) ...
If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.
... We now move on to the classes of perfectly divisible graphs, stable-perfect, and nice graphs. Problem 32 in [4] asks whether nice graphs can be recognized in polynomial time. The recognition problem for nice graphs turns out to be NP-complete. ...
We show that the following problems are NP-complete. 1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph? 2. Is the difference between the chromatic number and clique number at most 1 for every induced subgraph of a graph? 3. Can the vertex set of every induced subgraph of a graph be partitioned into two sets such that the first set induces a perfect graph, and the clique number of the graph induced by the second set is smaller than that of the original induced subgraph? 4. Does a graph contain a stable set whose deletion results in a perfect graph? The proofs of the NP-completeness of the four problems follow the same pattern: Showing that all the four problems are NP-complete when restricted to triangle-free graphs by using results of Maffray and Preissmann on 3-colorability and 4-colorability of triangle-free graphs
For a number , let denote the family of graphs which have girth and have no odd hole with length greater than . Wu, Xu and Xu conjectured that every graph in is 3‐colourable. Chudnovsky et al., Wu et al., and Chen showed that every graph in , and is 3‐colourable, respectively. In this paper, we prove that every graph in is 3‐colourable. This confirms Wu, Xu and Xu's conjecture.
Let G be a graph. We say that G is 2-divisible if for each induced subgraph H of G, either V(H) is a stable set, or V(H) can be partitioned into two sets A and B such that ω(H[A]) < ω(H) and ω(H[B]) < ω(H). A hole is an induced cycle of length at least 4, a bull is a graph consisting of a triangle with two disjoint pendant edges, a diamond is the graph obtained from K4 by removing an edge, a dart denotes the graph obtained from a diamond by adding a pendant edge to one vertex of degree 3, and a racket denotes the graph obtained from a diamond by adding a pendant edge to one vertex of degree 2. In this paper, we prove that every {odd hole, H}-free graph is 2-divisible, where H is a dart, or a racket, or a bull. As corollaries, χ(G)≤min{2ω(G)−1,(ω(G)+12)} if G is {odd hole, dart}-free, or {odd hole, racket}-free, or {odd hole, bull}-free.
If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the early 1980s, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.
A graph G with clique number ω(G) and chromatic number χ(G) is perfect if χ(H) = ω(H) for every induced subgraph H of G. A family G of graphs is called χ-bounded with binding function f if χ(G′) ≤ f(ω(G′)) holds whenever G∈ G and G′ is an induced subgraph of G. In this paper we will present a survey on polynomial χ-binding functions. Especially we will address perfect graphs, hereditary graphs satisfying the Vizing bound (χ≤ ω+ 1), graphs having linear χ-binding functions and graphs having non-linear polynomial χ-binding functions. Thereby we also survey polynomial χ-binding functions for several graph classes defined in terms of forbidden induced subgraphs, among them 2 K2-free graphs, Pk-free graphs, claw-free graphs, and diamond-free graphs.Families ofχ-bound graphs are natural candidates for polynomial approximation algorithms for the vertex coloring problem. (András Gyárfás [42]).
Polynomial algorithms for perfect graphs / M. Grötschel ; L. Lovász ; A. Schrijver. - In: Topics in perfect graphs / ed. by V. Chvátal ... - Amsterdam : North Holland Publ., 1984. - S. 325-356. - (Annals of discrete mathematics ; 21).
A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)|9). This is independent of the recent proof of the strong perfect graph conjecture.
We show that f(x)=⌊[Formulla presented]x⌋ is a θ-bounding function for the class of subcubic graphs and that it is best possible. This generalizes a result by Henning et al. (2012), who showed that θ(G)≤[Formulla presented]α(G) for any subcubic triangle-free graph G. Moreover, we provide a θ-bounding function for the class of K4-free graphs with maximum degree at most 4. Finally, we study the problem CLIQUE COVER for subclasses of planar graphs and graphs with bounded maximum degree: in particular, answering a question of Cerioli et al. (2008), we show it admits a PTAS for planar graphs.
A class of graphs is -bounded if there is a function f such that for every induced subgraph G of every graph in the class, where denote the chromatic number and clique number of G respectively. In 1987, Gy\'arf\'as conjectured that for every c, if is a class of graphs such that for every induced subgraph G of every graph in the class, then the class of complements of members of is -bounded. We prove this conjecture. Indeed, more generally, a class of graphs is -bounded if it has the property that no graph in the class has c+1 odd holes, pairwise disjoint and with no edges between them. The main tool is a lemma that if C is a shortest odd hole in a graph, and X is the set of vertices with at least five neighbours in V(C), then there is a three-vertex set that dominates X.
We prove a 1985 conjecture of Gyárfás that for all k, ℓ, every graph with sufficiently large chromatic number contains either a clique of cardinality more than k or an induced cycle of length more than ℓ.
The Erd\"os-Hajnal conjecture states that for every graph H, there exists a constant such that every graph G with no induced subgraph isomorphic to H has either a clique or a stable set of size at least . This paper is a survey of some of the known results on this conjecture.
An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyárfás made the conjecture that for all t there exists n such that every graph with no subgraph and no odd hole is n-colourable. We prove this conjecture.
The Erdös–Hajnal conjecture states that for every graph H, there exists a constant such that every graph G with no induced subgraph isomorphic to H has either a clique or a stable set of size at least . This article is a survey of some of the known results on this conjecture.
Abstract All K4-free graphs with no odd hole and no odd antihole are three-colourable, but what about K4- free graphs with no odd hole? They are not necessarily three-colourable, but we prove a conjecture of Ding that they are all four-colourable. This is a consequence of a decomposition theorem for such graphs; we prove that every such graph either has no odd antihole, or belongs to one of two explicitly-constructed classes, or admits a decomposition.
A graph is triangulated if it has no chordless cycle with four or more vertices. It follows that the complement of a triangulated graph cannot contain a chordless cycle with five or more vertices. We introduce a class of graphs (namely, weakly triangulated graphs) which includes both triangulated graphs and complements of triangulated graphs (we define a graph as weakly triangulated if neither it nor its complement contains a chordless cycle with five or more vertices). Our main result is a structural theorem which leads to a proof that weakly triangulated graphs are perfect.
In this paper we will consider Ramsey-type problems for finite graphs, r-partitions and hypergraphs. All these problems ask for the existence of large homogeneous (monochromatic) configurations of a certain kind under the condition that the size of the underlying set is large. As it is quite common in Ramsey theory, most of our results are not sharp and almost all of them lead to new problems which seem to be difficult. The problems we treat are only loosely connected. So we will state and explain them section by section.
We show that the question “Is a graph 3-colorable?” remains NP-complete when restricted to the class of triangle-free graphs with maximum degree 4. Likewise the question “Is a triangle-free graph k-colorable?” is shown to be NP-complete for any fixed value of k ⩾ 4.
A graph G is k-divisible if for each induced subgraph H of G with at least one edge, there is a partition of the vertex set of H into sets V1,…,Vk such that no Vi contains a maximum clique of H. We show that a claw-free graph is 2-divisible if and only if it does not contain an odd hole: we conjecture that this result is true for any graph, and present further conjectures relating 2-divisibility to the strong perfect graph conjecture. We also present related results involving the chromatic number and the stability number, with connections to Ramsey theory.
This paper presents an algorithm that determines whether an undirected graph has any holes of size at least k in O(nk−3M), where M is the time required to multiply two n by n matrices.
A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In this paper we prove that every even-hole-free graph has a bisimplicial vertex, which was originally conjectured by Reed.
Can the vertices of an arbitrary graph G be partitioned into A#B,sothatG[A] is a line-graph and G[B] is a forest? Can G be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if are additive induced-hereditary graph properties, then (P, Q)- colouring is NP-hard, with the sole exception of graph 2-colouring (the case where of finite edgeless graphs). Moreover, (P, NP-complete i# are both in NP. This completes the proof of a conjecture of Kratochvl and Schiermeyer, various authors having already settled many sub-cases.
On the divisibility of graphs, Discrete Math
- C T Hoang
- C Mcdiarmid
C. T. Hoang, C. McDiarmid, On the divisibility of graphs, Discrete Math. 242, 1?3 (2002),
145-156.