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Maria Chudnovsky

Publications and source records attributed to Maria Chudnovsky.

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arXiv2023

Induced subgraphs and tree-decompositions VII. Basic obstructions in $H$-free graphs

We say a class $\mathcal{C}$ of graphs is clean if for every positive integer $t$ there exists a positive integer $w(t)$ such that every graph in $\mathcal{C}$ with treewidth more than $w(t)$ contains an induced subgraph isomorphic to one of the following: the complete graph $K_t$, the complete bipartite graph $K_{t,t}$, a subdivision of the $(t\times t)$-wall or the line graph of a subdivision of the $(t \times t)$-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Weißauer) to prove that the class of all $H$-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph $H$) is clean if and only if $H$ is a forest whose components are subdivided stars. Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest $H$ as above, we show that forbidding certain connected graphs containing $H$ as an induced subgraph (rather than $H$ itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer $η$, a complete description of unavoidable connected induced subgraphs of a connected graph $G$ containing $η$ vertices from a suitably large given set of vertices in $G$. This is of independent interest, and will be used in subsequent papers in this series.

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arXiv2023

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all $k$ and $Δ$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a subdivision of the $(k\times k)$-wall or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For $t\geq 2$, a $t$-theta is a graph consisting of two nonadjacent vertices and three internally disjoint paths between them, each of length at least $t$. A $t$-pyramid is a graph consisting of a vertex $v$, a triangle $B$ disjoint from $v$ and three paths starting at $v$ and disjoint otherwise, each joining $v$ to a vertex of $B$, and each of length at least $t$. We prove that for all $k,t$ and $Δ$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a $t$-theta, or a $t$-pyramid, or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a $t$-theta for some $t\geq 2$). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every $Δ$ and subcubic subdivided caterpillar $T$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a subdivision of $T$ or the line graph of a subdivision of $T$ as an induced subgraph.

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arXiv2023

Graphs with no even holes and no sector wheels are the union of two chordal graphs

Sivaraman conjectured that if $G$ is a graph with no induced even cycle then there exist sets $X_1, X_2 \subseteq V(G)$ satisfying $V(G) = X_1 \cup X_2$ such that the induced graphs $G[X_1]$ and $G[X_2]$ are both chordal. We prove this conjecture in the special case where $G$ contains no sector wheel, namely, a pair $(H, w)$ where $H$ is an induced cycle of $G$ and $w$ is a vertex in $V(G) \setminus V(H)$ such that $N(w) \cap H$ is either $V(H)$ or a path with at least three vertices.

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arXiv2023

Characterizing and generalizing cycle completable graphs

The family of cycle completable graphs has several cryptomorphic descriptions, the equivalence of which has heretofore been proven by a laborious implication-cycle that detours through a motivating matrix completion problem. We give a concise proof, partially by introducing a new characterization. Then we generalize this family to ``$k$-quasichordal'' graphs, with three natural characterizations.

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arXiv2023

Pure pairs. X. Tournaments and the strong Erdos-Hajnal property

A pure pair in a tournament $G$ is an ordered pair $(A,B)$ of disjoint subsets of $V(G)$ such that every vertex in $B$ is adjacent from every vertex in $A$. Which tournaments $H$ have the property that if $G$ is a tournament not containing $H$ as a subtournament, and $|G|>1$, there is a pure pair $(A,B)$ in $G$ with $|A|,|B|\ge c|G|$, where $c>0$ is a constant independent of $G$? Let us say that such a tournament $H$ has the strong EH-property. As far as we know, it might be that a tournament $H$ has this property if and only if its vertex set has a linear ordering in which its backedges form a forest. Certainly this condition is necessary, but we are far from proving sufficiency. We make a small step in this direction, showing that if a tournament can be ordered with at most three backedges then it has the strong EH-property (except for one case, that we could not decide). In particular, every tournament with at most six vertices has the property, except for three that we could not decide. We also give a seven-vertex tournament that does not have the strong EH-property. This is related to the Erdos-Hajnal conjecture, which in one form says that for every tournament $H$ there exists $τ>0$ such that every tournament $G$ not containing $H$ as a subtournament has a transitive subtournament of cardinality at least $|G|^τ$. Let us say that a tournament $H$ satisfying this has the EH-property. It is known that every tournament with the strong EH-property also has the EH-property; so our result extends work by Berger, Choromanski and Chudnovsky, who proved that every tournament with at most six vertices has the EH-property, except for one that they did not decide.

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arXiv2023

Polynomial bounds for chromatic number VI. Adding a four-vertex path

A class of graphs is $χ$-bounded if there is a function $f$ such that every graph $G$ in the class has chromatic number at most $f(ω(G))$, where $ω(G)$ is the clique number of $G$; the class is polynomially $χ$-bounded if $f$ can be taken to be a polynomial. The Gyárfás-Sumner conjecture asserts that, for every forest $H$, the class of $H$-free graphs (graphs with no induced copy of $H$) is $χ$-bounded. Let us say a forest $H$ is good if it satisfies the stronger property that the class of $H$-free graphs is polynomially $χ$-bounded. Very few forests are known to be good: for example, it is open for the five-vertex path. Indeed, it is not even known that if every component of a forest $H$ is good then $H$ is good, and in particular, it was not known that the disjoint union of two four-vertex paths is good. Here we show the latter, and more generally, that if $H$ is good then so is the disjoint union of $H$ and a four-vertex path. We also prove a more general result: if every component of $H_1$ is good, and $H_2$ is any path (or broom) then the class of graphs that are both $H_1$-free and $H_2$-free is polynomially $χ$-bounded.

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arXiv2023

Complexity of $C_k$-coloring in hereditary classes of graphs

For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an \emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for every edge $uv\in E(G)$ it holds that $f(u)f(v)\in E(H)$. We are interested in the complexity of the problem $H$-{\sc Coloring}, which asks for the existence of an $H$-coloring of an input graph $G$. In particular, we consider $H$-{\sc Coloring} of $F$-free graphs, where $F$ is a fixed graph and $H$ is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-{\sc Coloring} of $P_t$-free graphs. We show that for every odd $k \geq 5$ the $C_k$-{\sc Coloring} problem, even in the list variant, can be solved in polynomial time in $P_9$-free graphs. The algorithm extends for the case of list version of $C_k$-{\sc Coloring}, where $k$ is an even number of length at least 10. On the other hand, we prove that if some component of $F$ is not a subgraph of a subdividecd claw, then the following problems are NP-complete in $F$-free graphs: a)extension version of $C_k$-{\sc Coloring} for every odd $k \geq 5$, b) list version of $C_k$-{\sc Coloring} for every even $k \geq 6$.

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arXiv2023

Cops and robbers on $P_5$-free graphs

We prove that every connected $P_5$-free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we first prove that every connected $P_5$-free graph $G$ with independence number at least three contains a three-vertex induced path with vertices $a \hbox{-} b \hbox{-} c$ in order, such that every neighbour of $c$ is also adjacent to one of $a,b$.

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arXiv2022

Induced subgraphs and tree decompositions IV. (Even hole, diamond, pyramid)-free graphs

A hole in a graph $G$ is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph $K_4$ by removing an edge. A pyramid is a graph consisting of a triangle called the base, a vertex called the apex, and three internally disjoint paths starting at the apex and disjoint otherwise, each joining the apex to a vertex of the base. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Cameron, da Silva, Huang, and Vušković proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. Sintiari and Trotignon provided a construction of (even hole, pyramid, $K_4$)-free graphs of arbitrarily large treewidth. Here, we show that for every $t$, (even hole, pyramid, diamond, $K_t$)-free graphs have bounded treewidth. The graphs constructed by Sintiari and Trotignon contain diamonds, so our result is sharp in the sense that it is false if we do not exclude diamonds. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses "non-crossing decompositions" methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of "non-crossing decompositions" to prove bounded treewidth in a graph class of unbounded maximum degree.

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arXiv2022

Non-uniform degrees and rainbow versions of the Caccetta-Häggkvist conjecture

The Caccetta-Häggkvist conjecture (denoted below CHC) states that the directed girth (the smallest length of a directed cycle) $dgirth(D)$ of a directed graph $D$ on $n$ vertices is at most $\lceil \frac{n}{δ^+(D)}\rceil$, where $δ^+(D)$ is the minimum out-degree of~$D$. We consider a version involving all out-degrees, not merely the minimum one, and prove that if $D$ does not contain a sink, then $dgirth(D) \le 2 \sum_{v\in V(D)} \frac{1}{deg^+(v)+1}$. In the spirit of a generalization of the CHC to rainbow cycles in \cite{ADH2019}, this suggests the conjecture that given non-empty sets $F_1, \ldots,F_n$ of edges of $K_n$, there exists a rainbow cycle of length at most $2\sum_{1\le i \le n}\frac{1}{|F_i|+1}$. We prove a bit stronger result when $1\le |F_i|\le 2$, thereby strengthening a result of DeVos et. al \cite{DDFGGHMM2021}. We prove a logarithmic bound on the rainbow girth in the case that the sets $F_i$ are triangles.

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arXiv2022

Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $c\log (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth. Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $G$ excluding the so-called three-path-configurations as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and Coloring admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.

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arXiv2022

Strengthening Rodl's theorem

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rodl showed in the 1980s that every H-free graph has large parts that are very dense or very sparse. More precisely, let us say that a graph F on n vertices is c-restricted if either F or its complement has maximum degree at most cn. Rodl proved that for every graph H, and every c>0, every H-free graph G has a linear-sized set of vertices inducing a c-restricted graph. We strengthen Rodl's result as follows: for every graph H, and all c>0, every H-free graph can be partitioned into a bounded number of subsets inducing c-restricted graphs.

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arXiv2022

Coloring Square-free Berge Graphs

We consider the class of Berge graphs that do not contain a chordless cycle of length $4$. We present a purely graph-theoretical algorithm that produces an optimal coloring in polynomial time for every graph in that class.

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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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