Strong Perfect Graph Conjecture Research Articles

Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P6, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P6, diamond)-free graph G is no larger than the maximum of 6 and the clique number of G. We do this by reducing the problem to imperfect (P6, diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical (P6, diamond, K6)-free graph. Together with the Lovász theta function, this gives a polynomial time algorithm to compute the chromatic number of (P6, diamond)-free graphs.

Forbidden Induced Pairs for Perfectness and $\omega$-Colourability of Graphs

We characterise the pairs of graphs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are perfect. Similarly, we characterise pairs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are $\omega$-colourable (that is, their chromatic number is equal to their clique number). More generally, we show characterizations of pairs $\{ X, Y \}$ for perfectness and $\omega$-colourability of all connected $\{ X, Y \}$-free graphs which are of independence at least~$3$, distinct from an odd cycle, and of order at least $n_0$, and similar characterisations subject to each subset of these additional constraints. (The classes are non-hereditary and the characterisations for perfectness and $\omega$-colourability are different.) We build on recent results of Brause et al. on $\{ K_{1,3}, Y \}$-free graphs, and we use Ramsey's Theorem and the Strong Perfect Graph Theorem as main tools. We relate the present characterisations to known results on forbidden pairs for $\chi$-boundedness and deciding $k$-colourability in polynomial time.

Point Partition Numbers: Perfect Graphs

Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $$t\ge 1$$ , denote by $$\mathcal{MG}_t$$ the class of graphs whose maximum multiplicity is at most t. A graph G is called strictly t-degenerate if every non-empty subgraph H of G contains a vertex v whose degree in H is at most $$t-1$$ . The point partition number $$\chi _t(G)$$ of G is the smallest number of colors needed to color the vertices of G so that each vertex receives a color and vertices with the same color induce a strictly t-degenerate subgraph of G. So $$\chi _1$$ is the chromatic number, and $$\chi _2$$ is known as the point aboricity. The point partition number $$\chi _t$$ with $$t\ge 1$$ was introduced by Lick and White (Can J Math 22:1082–1096, 1970). If H is a simple graph, then tH denotes the graph obtained from H by replacing each edge of H by t parallel edges. Then $$\omega _t(G)$$ is the largest integer n such that G contains a $$tK_n$$ as a subgraph. Let G be a graph belonging to $$\mathcal{MG}_t$$ . Then $$\omega _t(G)\le \chi _t(G)$$ and we say that G is $$\chi _t$$ -perfect if every induced subgraph H of G satisfies $$\omega _t(H)=\chi _t(H)$$ . Based on the Strong Perfect Graph Theorem due to Chudnowsky, Robertson, Seymour and Thomas (Ann Math 164:51–229, 2006), we give a characterization of $$\chi _t$$ -perfect graphs of $$\mathcal{MG}_t$$ by a set of forbidden induced subgraphs (see Theorems 2 and 3). We also discuss some complexity problems for the class of $$\chi _t$$ -critical graphs.

An optimal χ‐bound for (P6, diamond)‐free graphs

AbstractIn this paper we show that every (, diamond)‐free graph satisfies , where and are the chromatic number and clique number of , respectively. Our bound is attained by the complement of the famous 27‐vertex Schläfli graph. Our result unifies previously known results on the existence of linear ‐binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect (, diamond)‐free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well‐known characterization of 3‐colourable ‐free graphs.

The Normal Graph Conjecture for Two Classes of Sparse Graphs

Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover $$\mathcal {Q}$$ and a stable set cover $$\mathcal {S}$$ s.t. every clique in $$\mathcal {Q}$$ intersects every stable set in $$\mathcal {S}$$ . Normal graphs can be considered as closure of perfect graphs by means of co-normal products and graph entropy. Perfect graphs have been characterized as those graphs without odd holes and odd antiholes as induced subgraphs (Strong Perfect Graph Theorem, Chudnovsky et al.). Korner and de Simone observed that $$C_5$$ , $$C_7$$ , and $$\overline{C}_7$$ are minimal not normal and conjectured, in analogy to the Strong Perfect Graph Theorem, that every $$(C_5, C_7, \overline{C}_7)$$ -free graph is normal (Normal Graph Conjecture, Korner and de Simone). Recently, this conjecture has been disproved by Harutyunyan, Pastor and Thomasse. However, in this paper we verify it for two classes of sparse graphs, 1-trees and cacti. In addition, we provide both linear time recognition algorithms and characterizations for the normal graphs within these two classes.

A short proof of the wonderful lemma

AbstractThe Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.

Counting Homomorphisms to Square-Free Graphs, Modulo 2

We study the problem ⊕H oms T o H of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph H . A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any H that contains no 4-cycles, ⊕H oms T o H is either in polynomial time or is ⊕P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.

Clique–Stable Set separation in perfect graphs with no balanced skew-partitions

Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique–Stable Set separation in a non-hereditary subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates a clique K and a stable set S if K⊆B and S⊆W. A Clique–Stable Set separator is a family of cuts such that for every clique K, and for every stable set S disjoint from K, there exists a cut in the family that separates K and S. Given a class of graphs, the question is to know whether every graph of the class admits a Clique–Stable Set separator containing only polynomially many cuts. It was recently proved to be false for the class of all graphs (Göös, 2015), but it remains open for perfect graphs, which was Yannakakis’ original question. Here we investigate this problem on perfect graphs with no balanced skew-partition; the balanced skew-partition was introduced in the decomposition theorem of Berge graphs which led to the celebrated proof of the Strong Perfect Graph Theorem. Recently, Chudnovsky, Trotignon, Trunck and Vušković proved that forbidding this unfriendly decomposition permits to recursively decompose Berge graphs (more precisely, Berge trigraphs) using 2-join and complement 2-join until reaching a “basic” graph, and in this way, they found an efficient combinatorial algorithm to color those graphs.We apply their decomposition result to prove that perfect graphs with no balanced skew-partition admit a quadratic-size Clique–Stable Set separator, by taking advantage of the good behavior of 2-join with respect to this property. We then generalize this result and prove that the Strong Erdős–Hajnal property holds in this class, which means that every such graph has a linear-size biclique or complement biclique. This is remarkable since the property does not hold for all perfect graphs (Fox, 2006), and this is motivated here by the following statement: when the Strong Erdős–Hajnal property holds in a hereditary class of graphs, then both the Erdős–Hajnal property and the polynomial Clique–Stable Set separation hold. Finally, we define the generalized k-join and generalize both our results on classes of graphs admitting such a decomposition.

Perfect Graphs with No Balanced Skew‐Partition are 2‐Clique‐Colorable

AbstractA graph G is perfect if for all induced subgraphs H of G, . A graph G is Berge if neither G nor its complement contains an induced odd cycle of length at least five. The Strong Perfect Graph Theorem [9] states that a graph is perfect if and only if it is Berge. The Strong Perfect Graph Theorem was obtained as a consequence of a decomposition theorem for Berge graphs [M. Chudnovsky, Berge trigraphs and their applications, PhD thesis, Princeton University, 2003; M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann Math 164 (2006), 51–229.], and one of the decompositions in this decomposition theorem was the “balanced skew‐partition.” A clique‐coloring of a graph G is an assignment of colors to the vertices of G in such a way that no inclusion‐wise maximal clique of G of size at least two is monochromatic, and the clique‐chromatic number of G, denoted by , is the smallest number of colors needed to clique‐color G. There exist graphs of arbitrarily large clique‐chromatic number, but it is not known whether the clique‐chromatic number of perfect graphs is bounded. In this article, we prove that every perfect graph that does not admit a balanced skew‐partition is 2‐clique colorable. The main tool used in the proof is a decomposition theorem for “tame Berge trigraphs” due to Chudnovsky et al. (http://arxiv.org/abs/1308.6444).

Perfect Digraphs

AbstractThe clique number of a digraph D is the size of the largest bidirectionally complete subdigraph of D. D is perfect if, for any induced subdigraph H of D, the dichromatic number defined by Neumann‐Lara (The dichromatic number of a digraph, J. Combin. Theory Ser. B 33 (1982), 265–270) equals the clique number . Using the Strong Perfect Graph Theorem (M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006), 51–229) we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Modifying a recent proof of Bang‐Jensen et al. (Finding an induced subdivision of a digraph, Theoret. Comput. Sci. 443 (2012), 10–24) we show that the recognition of perfect digraphs is co‐‐complete. It turns out that perfect digraphs are exactly the complements of clique‐acyclic superorientations of perfect graphs. Thus, we obtain as a corollary that complements of perfect digraphs have a kernel, using a result of Boros and Gurvich (Perfect graphs are kernel solvable, Discrete Math. 159 (1996), 35–55). Finally, we prove that it is ‐complete to decide whether a perfect digraph has a kernel.

Envy-free pricing in multi-item markets

In this article, we study revenue maximizing envy-free pricing in multi-item markets: There are m indivisible items with unit supply each and n potential buyers where each buyer is interested in acquiring one item. The goal is to determine allocations (a matching between buyers and items) and prices of all items to maximize total revenue given that all buyers are envy-free. We give a polynomial time algorithm to compute a revenue maximizing envy-free pricing when every buyer evaluates at most two items at a positive valuation, by reducing it to an instance of weighted independent set in a perfect graph and applying the Strong Perfect Graph Theorem. We complement our result by showing that the problem becomes NP-hard if some buyers are interested in at least three items.

Fast recognition of doubled graphs

Double split graphs form one of the five basic classes in the proof of the strong perfect graph theorem by Chudnovsky et al. [3]. A doubled graph is any induced subgraph of a double split graph. We show here that deciding if a graph G is a doubled graph can be done in time O(|V(G)|+|E(G)|) by analyzing the degree structure of the graph.

Packing and covering with linear programming: A survey

This paper considers the polyhedral results and the min–max results on packing and covering problems of the decade. Since the strong perfect graph theorem (published in 2006), the main such results are available for the packing problem, however there are still important polyhedral questions that remain open. For the covering problem, the main questions are still open, although there has been important progress. We survey some of the main results with emphasis on those where linear programming and graph theory come together. They mainly concern the covering of cycles or dicycles in graphs or signed graphs, either with vertices or edges; this includes the multicut and integral multiflow problems.

The chromatic gap and its extremes

The chromatic gap is the difference between the chromatic number and the clique number of a graph. Here we investigate gap(n), the maximum chromatic gap over graphs on n vertices. Can the extremal graphs be explored? While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey-theory and matching theory leads to a simple and (almost) exact formula for gap(n) in terms of Ramsey-numbers. For our purposes it is more convenient to work with the covering gap, the difference between the clique cover number and stability number of a graph and this is what we call the gap of a graph. Notice that the well-studied family of perfect graphs are the graphs whose induced subgraphs have gap zero. The maximum of the (covering) gap and the chromatic gap running on all induced subgraphs will be called perfectness gap.Using α(G) for the cardinality of a largest stable (independent) set of a graph G, we define α(n)=minα(G) where the minimum is taken over triangle-free graphs on n vertices. It is easy to observe that α(n) is essentially an inverse Ramsey function, defined by the relation R(3,α(n))⩽n<R(3,α(n)+1). Our main result is that gap(n)=⌈n/2⌉−α(n), possibly with the exception of small intervals (of length at most 15) around the Ramsey-numbers R(3,m), where the error is at most 3.The central notions in our investigations are the gap-critical and the gap-extremal graphs. A graph G is gap-critical if for every proper induced subgraph H⊂G, gap(H)<gap(G) and gap-extremal if it is gap-critical with as few vertices as possible (among gap-critical graphs of the same gap). The strong perfect graph theorem, solving a long standing conjecture of Berge that stimulated a broad area of research, states that gap-critical graphs with gap 1 are the holes (chordless odd cycles of length at least five) and antiholes (complements of holes). The next step, the complete description of gap-critical graphs with gap 2 would probably be a very difficult task. As a very first step, we prove that there is a unique 2-extremal graph, 2C5, the union of two disjoint (chordless) cycles of length five.In general, for t⩾0, we denote by s(t) the smallest order of a graph with gap t and we call a graph is t-extremal if it has gap t and order s(t). Equivalently, s(t) is the smallest order of a graph with perfectness gap equal to t. It is tempting to conjecture that s(t)=5t with equality for the graph tC5. However, for t⩾3 the graph tC5 has gap t but it is not gap-extremal (although gap-critical). We shall prove that s(3)=13, s(4)=17 and s(5)∈{20,21}. Somewhat surprisingly, after the uncertain values s(6)∈{23,24,25}, s(7)∈{26,27,28}, s(8)∈{29,30,31}, s(9)∈{32,33} we can show that s(10)=35. On the other hand we can easily show that s(t) is asymptotically equal to 2t, that is, gap(n) is asymptotic to n/2. According to our main result the gap is actually equal to ⌈n/2⌉−α(n), unless n is in an interval [R,R+14] where R is a Ramsey-number, and if this exception occurs the gap may be larger than this value by only a small constant (at most 3).Our study provides some new properties of Ramsey graphs them selves: it shows that triangle-free Ramsey graphs have high matchability and connectivity properties, leading possibly to new bounds on Ramsey-numbers.

A linear time algorithm for the induced disjoint paths problem in planar graphs

In this paper, we consider a problem which we call the induced disjoint paths problem (IDPP) for planar graphs. We are given a planar graph G and a collection of vertex pairs { ( s 1 , t 1 ) , … , ( s k , t k ) } . The objective is to find k paths P 1 , … , P k such that P i is a path from s i to t i and P i and P j have neither common vertices nor adjacent vertices for any distinct i , j . This problem setting is a generalization of the disjoint paths problem, since if we subdivide each edge, then desired disjoint paths would be induced disjoint paths. The problem is motivated by not only the disjoint paths problem but also the recognition of an induced subgraph. The latter has been developed in recent years, and this is actually connected to the Strong Perfect Graph Theorem (Chudnovsky, et al., 2006) [1], and the recognition of the perfect graphs (Chudnovsky, et al., 2005) [2]. Our main result in this paper is to give a linear time algorithm for the IDPP for planar graphs. This generalizes the result by Reed, Robertson, Schrijver and Seymour (1993) [14]. This also gives a polynomial time algorithm to find an induced circuit through given k vertices in planar graphs if one exists when k is fixed. The case k = 2 was previously proved by McDiarmid, Reed, Schrijver and Shepherd (1994) [10].

On Roussel–Rubio-type lemmas and their consequences

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward’s decomposition theorem for weakly chordal graphs, relying on a Roussel–Rubio-type lemma. We recall how Roussel–Rubio-type lemmas yield very short proofs of the existence of even pairs in weakly chordal graphs and Meyniel graphs.

Optimal Envy-Free Pricing with Metric Substitutability

We study the unit-demand envy-free pricing problem faced by a profit-maximizing seller with unlimited supply when there is metric substitutability among the items—consumer $i$'s value for item $j$ is $v_i-c_{i,j}$, and the substitution costs, $\{c_{i,j}\}$, form a metric. Our model is motivated by the observation that sellers often sell the same product at different prices in different locations, and rational consumers optimize the tradeoff between prices and substitution costs. While the general envy-free pricing problem is hard to approximate, we show that the problem of maximizing revenue with metric substitutability among items can be solved exactly in polynomial time. We do this by first showing that in any optimal price vector, the set of nodes that pay exactly their value uniquely determines which nodes buy an item and what price they pay, and therefore the revenue. We transform the problem of finding an optimal set of such nodes to an instance of weighted independent set on a perfect graph which can be solved in polynomial time by the strong perfect graph theorem, proving the result. We then analyze the computational tractability of various extensions to our model. We begin with relaxing the metric substitutability requirement and show that when the substitution costs do not form a metric, even if a $(1+\epsilon)$-approximate triangle inequality holds, the problem becomes NP-hard. Thus the triangle inequality characterizes the threshold at which the problem goes from “tractable” to “hard.” We then relax assumptions on the supply and demand. We consider restricting supplies to a subset of locations, or the amount of supplies, or allowing buyers to demand more than one unit. In all cases, the problem becomes NP-hard. In addition, the multiunit demand case illustrates an interesting paradoxical nonmonotonicity: The optimal revenue the seller can extract can actually decrease when consumers' demands increase. We show the revenue maximization problem with multiunit demand is APX-hard even for the simplest valuations with equal marginal values for all items up to the demand constraint, and demands of at most 3.

Forbidden Induced Subgraphs of Double-split Graphs

In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic classes are closed under the operation of taking induced subgraphs (and have known forbidden subgraph characterizations), while the fifth one, consisting of double-split graphs, is not. A graph is doubled if it is an induced subgraph of a double-split graph. We find the forbidden induced subgraph characterization of doubled graphs; it contains 44 graphs.

Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J ( H ) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R / J s , for any power J s of J, in terms of the coloring properties of hypergraphs arising from H . We also give an algebraic method for determining the chromatic number of H , proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely algebraic characterizations of perfect graphs, independent of the Strong Perfect Graph Theorem; the first characterization is in terms of the sets Ass ( R / J s ) , while the second characterization is in terms of the saturated chain condition for associated primes.

Algorithms for finding an induced cycle in planar graphs

In this paper, we consider the problem of finding an induced cycle passing through k given vertices, which we call the induced cycle problem. The significance of finding induced cycles stems from the fact that a precise characterization of perfect graphs would require understanding the structure of graphs without an odd induced cycle and its complement. There has been huge progress in the recent years, especially, the Strong Perfect Graph Conjecture was solved in [6]. Concerning recognition of perfect graphs, there had been a long-standing open problem for detecting an odd hole and its complement, and finally this was solved in [4]. Unfortunately, the problem of finding an induced cycle passing through two given vertices is NP-complete in a general graph [2]. However, if the input graph is constrained to be planar and k is fixed, then the induced cycle problem can be solved in polynomial time [11]–[13]. In particular, an O(n 2) time algorithm is given for the case k=2 by McDiarmid, Reed, Schrijver and Shepherd [14], where n is the number of vertices of the input graph. Our main results in this paper are to improve their result in the following sense. The number of vertices k is allowed to be non-trivially super constant number, up to $$k = o\left( {\left( {\tfrac{{\log n}}{{\log \log n}}} \right)^{\tfrac{2}{3}} } \right)$$. More precisely, when $$k = o\left( {\left( {\tfrac{{\log n}}{{\log \log n}}} \right)^{\tfrac{2}{3}} } \right)$$, then the ICP can be solved in O(n 2+ɛ) time for any ɛ > 0. The time complexity is linear if k is fixed. We note that the linear time algorithm (the second result) is independent from the first result. Let us observe that if k is as a part of the input, then the problem is still NP-complete, and so we need to impose some condition on k.