arXiv subjects

Maria Chudnovsky

Publications and source records attributed to Maria Chudnovsky.

At least 91 records · Page 5Linked to original sources

arXiv2022

Induced subgraphs and tree decompositions V. One neighbor in a hole

What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the "basic treewidth obstructions"). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so-called "layered wheels," contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.

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arXiv2022

Induced subgraphs and tree decompositions V. Small components of big vertices

Aboulker, Adler, Kim, Sintiari, and Trotignon conjectured that every graph with bounded maximum degree and large treewidth must contain, as an induced subgraph, a large subdivided wall, or the line graph of a large subdivided wall. This conjecture was recently proved by Korhonen, but the problem of identifying the obstacles to bounded treewidth in the general case (that is, without the bounded maximum degree condition) remains wide open. Examples of structures of large treewidth which avoid the "usual suspects" have been constructed by Sintiari and Trotignon, and by Davies. In this note, we aim to better isolate the features of these examples that lead to large treewidth. To this end, we prove the following result. Let $G$ be a graph, and write $γ(G)$ for the size of a largest connected component in the graph induced by $G$ on the set of vertices of degree at least 3. If $γ(G)$ is small and the treewidth of $G$ is large, then $G$ must contain a large subdivided wall or the line graph of a large subdivided wall. This result is the best possible, in the sense that the conclusion fails if we replace 3 by any larger number in the definition of $γ(G)$, as evidenced by Davies' example.

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arXiv2022

Bipartite graphs with no $K_6$ minor

A theorem of Mader shows that every graph with average degree at least eight has a $K_6$ minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have $K_6$ minors, but minimum degree six is certainly not enough. For every $c>0$ there are arbitrarily large graphs with average degree at least $8-c$ and minimum degree at least six, with no $K_6$ minor. But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every $c>0$ there are arbitrarily large bipartite graphs with average degree at least $8-c$ and no $K_6$ minor. But surprisingly, going to minimum degree now makes a significant difference. We will show that every bipartite graph with minimum degree at least six has a $K_6$ minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.

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arXiv2022

Forbidden induced pairs for perfectness and $ω$-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 $ω$-colourable (that is, their chromatic number is equal to their clique number). More generally, we show characterizations of pairs $\{ X, Y \}$ for perfectness and $ω$-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 $ω$-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 $χ$-boundedness and deciding $k$-colourability in polynomial time.

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arXiv2022

Stable sets in flag spheres

We provide lower and upper bounds on the minimum size of a maximum stable set over graphs of flag spheres, as a function of the dimension of the sphere and the number of vertices. Further, we use stable sets to obtain an improved Lower Bound Theorem for the face numbers of flag spheres.

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arXiv2022

Polynomial bounds for chromatic number VII. Disjoint holes

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to its clique number. In this paper we show that, for any $k$, if $G$ does not contain a $k$-multihole, then its chromatic number is at most a polynomial function of its clique number. We show that the same result holds if we ask for all the holes to be odd or of length four; and if we ask for the holes to be longer than any fixed constant or of length four. This is part of a broader study of graph classes that are polynomially $χ$-bounded.

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arXiv2022

Polynomial-time algorithm for Maximum Independent Set in bounded-degree graphs with no long induced claws

For graphs $G$ and $H$, we say that $G$ is $H$-free if it does not contain $H$ as an induced subgraph. Already in the early 1980s Alekseev observed that if $H$ is connected, then the \textsc{Max Weight Independent Set} problem (MWIS) remains \textsc{NP}-hard in $H$-free graphs, unless $H$ is a path or a subdivided claw, i.e., a graph obtained from the three-leaf star by subdividing each edge some number of times (possibly zero). Since then determining the complexity of MWIS in these remaining cases is one of the most important problems in algorithmic graph theory. A general belief is that the problem is polynomial-time solvable, which is witnessed by algorithmic results for graphs excluding some small paths or subdivided claws. A more conclusive evidence was given by the recent breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that MWIS can be solved in quasipolynomial time in $H$-free graphs, where $H$ is any fixed path. If $H$ is an arbitrary subdivided claw, we know much less: The problem admits a QPTAS and a subexponential-time algorithm [Chudnovsky et al., SODA 2019]. In this paper we make an important step towards solving the problem by showing that for any subdivided claw $H$, MWIS is polynomial-time solvable in $H$-free graphs of bounded degree.

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arXiv2022

Induced subgraphs and tree decompositions I. Even-hole-free graphs of bounded degree

Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph has treewidth $k$ if it can be decomposed by a sequence of noncrossing cutsets of size at most $k$ into pieces of size at most $k+1$. The study of hereditary graph classes (i.e. those closed under vertex deletion only) reveals a different picture, where cutsets that are not necessarily bounded in size (such as star cutsets, 2-joins and their generalization) are required to decompose the graph into simpler pieces that are structured but not necessarily bounded in size. A number of such decomposition theorems are known for complex hereditary graph classes, including even-hole-free graphs, perfect graphs and others. These theorems do not describe the global structure in the sense that a tree decomposition does, since the cutsets guaranteed by them are far from being noncrossing. They are also of limited use in algorithmic applications. We show that in the case of even-hole-free graphs of bounded degree the cutsets described in the previous paragraph can be partitioned into a bounded number of well-behaved collections. This allows us to prove that even-hole-free graphs with bounded degree have bounded treewidth, resolving a conjecture of Aboulker, Adler, Kim, Sintiari and Trotignon [arXiv:2008.05504]. As a consequence, it follows that many algorithmic problems can be solved in polynomial time for this class, and that even-hole-freeness is testable in the bounded degree graph model of property testing. In fact we prove our results for a larger class of graphs, namely the class of $C_4$-free odd-signable graphs with bounded degree.

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arXiv2022

Proof of a conjecture of Plummer and Zha

Say a graph $G$ is a {\em pentagraph} if every cycle has length at least five, and every induced cycle of odd length has length five. N. Robertson proposed the conjecture that the Petersen graph is the only pentagraph that is three-connected and internally 4-connected, but this was disproved by M. Plummer and X. Zha in 2014. Plummer and Zha conjectured that every 3-connected, internally 4-connected pentagraph is three-colourable. We prove this: indeed, we will prove that every pentagraph is three-colourable.

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arXiv2021

Graphs with polynomially many minimal separators

We show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separators if only three of the four induced subgraphs are excluded. As a consequence, there is a polynomial time algorithm to solve the maximum weight independent set problem for the class of (theta, pyramid, prism, turtle)-free graphs. Since every prism, theta, and turtle contains an even hole, this also implies a polynomial time algorithm to solve the maximum weight independent set problem for the class of (pyramid, even hole)-free graphs.

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arXiv2021

Rainbow paths and large rainbow matchings

A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\frac{2}{3}n-1$, improving on the trivial $\frac{1}{2}n$, and an analogous result for hypergraphs. For $\{C_3,C_5\}$-free graphs and for disjoint matchings we obtain a lower bound of $\frac{3n}{4}-O(1)$. We also discuss a conjecture on rainbow alternating paths, that if true would yield a lower bound of $n-\sqrt{2n}$. We prove the non-alternating (ordinary paths) version of this conjecture.

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arXiv2021

A note on simplicial cliques

Motivated by an application in condensed matter physics and quantum information theory, we prove that every non-null even-hole-free claw-free graph has a simplicial clique, that is, a clique $K$ such that for every vertex $v \in K$, the set of neighbours of $v$ outside of $K$ is a clique. In fact, we prove the existence of a simplicial clique in a more general class of graphs defined by forbidden induced subgraphs.

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arXiv2021

Tournaments and the Strong Erdős-Hajnal Property

A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős-Hajnal Conjecture, states that for every tournament $S$ there exists $ε(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $n^{ε(S)}$ vertices. Let $C_5$ be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon-Pach-Solymosi conjecture is known to be true for the case when $S=C_5$. Here we prove a strengthening of this result, showing that in every tournament $T$ with no subtorunament isomorphic to $C_5$ there exist disjoint vertex subsets $A$ and $B$, each containing a linear proportion of the vertices of $T$, and such that every vertex of $A$ is adjacent to every vertex of $B$.

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arXiv2021

Induced subgraphs of graphs with large chromatic number. V. Chandeliers and strings

It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph, a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, forests and triangles are pervasive, where H is pervasive (in some class of graphs) if for all s>0, every graph in the class with bounded clique number and sufficiently large chromatic number contains an induced subdivision of H, with every edge subdivided at least s times. Which other graphs are pervasive? Chalopin, Esperet, Li and Ossona de Mendez proved that every such graph is a forest of lanterns: roughly, the blocks are lanterns (graphs obtained from a tree by adding one extra vertex), and there are rules about how blocks fit together. It is not known whether every forest of lanterns is pervasive; but in another paper two of us prove that banana trees (multigraphs obtained from a forest by adding parallel edges) are pervasive, thus generalizing the two results above. This paper contains the first half of the proof, which works for any forest of lanterns, not just for banana trees. A class of graphs is r-controlled if for every graph in the class, its chromatic number is at most some function (determined by the class) of the largest chromatic number of an r-ball in the graph. In this paper we prove that for all r>1, every forest of lanterns is pervasive in every r-controlled class These results turn out particularly nicely when applied to string graphs (intersection graphs of sets of curves in the plane). A chandelier is a graph obtained from a tree by adding a vertex adjacent to its leaves. We prove that the class of string graphs is 2-controlled, and thus forests of lanterns are pervasive in this class. Furthermore, string graphs of sufficiently large chromatic number and bounded clique number contain any fixed chandelier as an induced subgraph.

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arXiv2021

Erdos-Hajnal for graphs with no 5-hole

The Erdos-Hajnal conjecture says that for every graph H there exists c>0 such that every graph G not containing H as an induced subgraph has a clique or stable set of cardinality at least |G|^c. We prove that this is true when H is a cycle of length five. We also prove several further results: for instance, that if C is a cycle and H is the complement of a forest, there exists c>0 such that every graph G containing neither of C,H as an induced subgraph has a clique or stable set of cardinality at least |G|^c.

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arXiv2020

Sparse graphs with no polynomial-sized anticomplete pairs

A graph is "$H$-free" if it has no induced subgraph isomorphic to $H$. A conjecture of Conlon, Fox and Sudakov states that for every graph $H$, there exists $s>0$ such that in every $H$-free graph with $n>1$ vertices, either some vertex has degree at least $sn$, or there are two disjoint sets of vertices, of sizes at least $sn^s$ and $sn$, anticomplete to each other. We prove this holds for a large class of graphs $H$, and we prove that something like it holds for all graphs $H$. Say $H$ is "almost-bipartite" if $H$ is triangle-free and $V(H)$ can be partitioned into a stable set and a set inducing a graph of maximum degree at most one. We prove that the conjecture above holds for when $H$ is almost-bipartite. We also prove a stronger version where instead of excluding $H$ we restrict the number of copies of $H$. We prove some variations on the conjecture, such as: for every graph $H$, there exists $s >0$ such that in every $H$-free graph with $n>1$ vertices, either some vertex has degree at least $sn$, or there are two disjoint sets $A, B$ of vertices with $|A||B| > s n^{1 + s}$, anticomplete to each other.

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arXiv2020

Concatenating bipartite graphs

Let $x,y\in(0,1]$ and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbours in $B$, and every vertex in $B$ has at least $y|C|$ neighbours in $C$. We denote by $ϕ(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v$ in $C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. The function $ϕ$ is interesting, and we investigate some of its properties. For instance, we show that it is symmetric in $x$ and $y$, and that it has a discontinuity at $x=y=1/k$ for all integers $k>1$. We raise a number of questions and conjectures.

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arXiv2020

Strongly Perfect Claw-free Graphs -- A Short Proof

A graph is strongly perfect if every induced subgraph H has a stable set that meets every maximal clique of H. A graph is claw-free if no vertex has three pairwise non-adjacent neighbors. The characterization of claw-free graphs that are strongly perfect by a set of forbidden induced subgraphs was conjectured by Ravindra in 1990 and was proved by Wang in 2006. Here we give a shorter proof of this characterization.

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