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Sophie Spirkl

Publications and source records attributed to Sophie Spirkl.

At least 55 records · Page 3Linked to original sources
arXiv2024

Pure pairs. IX. Transversal trees

Fix k>0, and let G be a graph, with vertex set partitioned into k subsets (`blocks') of approximately equal size. An induced subgraph of G is transversal (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly k vertices). A pure pair in G is a pair X,Y of disjoint subsets of V(G) such that either all edges between X,Y are present or none are; and in the present context we are interested in pure pairs (X,Y) where each of X,Y is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.

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arXiv2024

Hitting all maximum stable sets in $P_5$-free graphs

We prove that every $P_5$-free graph of bounded clique number contains a small hitting set of all its maximum stable sets. More generally, let us say a class $\mathcal{C}$ of graphs is $η$-bounded if there exists a function $h:\mathbb{N}\rightarrow \mathbb{N}$ such that $η(G)\leq h(ω(G))$ for every graph $G\in \mathcal{C}$, where $η(G)$ denotes smallest cardinality of a hitting set of all maximum stable sets in $G$, and $ω(G)$ is the clique number of $G$. Also, $\mathcal{C}$ is said to be polynomially $η$-bounded if in addition $h$ can be chosen to be a polynomial. We introduce $η$-boundedness inspired by a question of Alon and motivated by a number of meaningful similarities to $χ$-boundedness. In particular, we propose an analogue of the Gyárfás-Sumner conjecture, that the class of all $H$-free graphs is $η$-bounded if (and only if) $H$ is a forest. Like $χ$-boundedness, the case where $H$ is a star is easy to verify, and we prove two non-trivial extensions of this: $H$-free graphs are $η$-bounded if (1) $H$ has a vertex incident with all edges of $H$, or (2) $H$ can be obtained from a star by subdividing at most one edge, exactly once. Unlike $χ$-boundedness, the case where $H$ is a path is surprisingly hard. Our main result mentioned at the beginning shows that $P_5$-free graphs are $η$-bounded. The proof is rather involved compared to the classical ``Gyárfás path'' argument which establishes, for all $t$, the $χ$-boundedness of $P_t$-free graphs. It remains open whether $P_t$-free graphs are $η$-bounded for $t\geq 6$. It also remains open whether $P_5$-free graphs are polynomially $η$-bounded, which, if true, would imply the Erdős-Hajnal conjecture for $P_5$-free graphs. But we prove that $H$-free graphs are polynomially $η$-bounded if $H$ is a proper induced subgraph of $P_5$.

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arXiv2024

On prime Cayley graphs

The decomposition of complex networks into smaller, interconnected components is a central challenge in network theory with a wide range of potential applications. In this paper, we utilize tools from group theory and ring theory to study this problem when the network is a Cayley graph. In particular, we answer the following question: Which Cayley graphs are prime?

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arXiv2023

List-$k$-Coloring $H$-free graphs for all $k>4$

Given an integer $k>4$ and a graph $H$, we prove that, assuming P$\neq$NP, the List-$k$-Coloring Problem restricted to $H$-free graphs can be solved in polynomial time if and only if either every component of $H$ is a path on at most three vertices, or removing the isolated vertices of $H$ leaves an induced subgraph of the five-vertex path. In fact, the "if" implication holds for all $k\geq 1$.

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arXiv2023

Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole

A clock is a graph consisting of an induced cycle $C$ and a vertex not in $C$ with at least two non-adjacent neighbours in $C$. We show that every clock-free graph of large treewidth contains a "basic obstruction" of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall.

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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

Pure pairs. VIII. Excluding a sparse graph

A pure pair of size $t$ in a graph $G$ is a pair $A,B$ of disjoint sets of $t$ vertices such that $A$ is either complete or anticomplete to $B$. It is known that, for every forest $H$, every graph on $n\ge2$ vertices that does not contain $H$ or its complement as an induced subgraph has a pure pair of size $Ω(n)$; furthermore, this only holds when $H$ or its complement is a forest. In this paper, we look at pure pairs of size $n^{1-c}$, where $0 c$.

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arXiv2023

Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture

For a digraph $G$ and $v \in V(G)$, let $δ^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-Häggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $δ^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size at least $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. Let us call $(α, β)$ \emph{triangular} if every simple edge-colored graph on $n$ vertices with at least $αn$ color classes, each with at least $βn$ edges, has a rainbow triangle. Aharoni, Holzman, and DeVos showed the following: $(9/8,1/3)$ is triangular; $(1,2/5)$ is triangular. In this paper, we improve those bounds, showing the following: $(1.1077,1/3)$ is triangular; $(1,0.3988)$ is triangular. Our methods give results for infinitely many pairs $(α, β)$, including $β< 1/3$; we show that $(1.3481,1/4)$ is triangular.

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arXiv2023

Further approximations for Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture

For a digraph $G$ and $v \in V(G)$, let $δ^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-Häggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $δ^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. With Pelikánová and Pokorná, we showed that this conjecture is true if each color class has size $Ω(k\log k)$. In this paper, we present a proof of the conjecture if each color class has size $Ω(k)$, which improved the previous result and is only a constant factor away from Aharoni's conjecture. We also consider what happens when the condition on the number of colors is relaxed.

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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

On heroes in digraphs with forbidden induced forests

We continue a line of research which studies which hereditary families of digraphs have bounded dichromatic number. For a class of digraphs $\mathcal{C}$, a hero in $\mathcal{C}$ is any digraph $H$ such that $H$-free digraphs in $\mathcal{C}$ have bounded dichromatic number. We show that if $F$ is an oriented star of degree at least five, the only heroes for the class of $F$-free digraphs are transitive tournaments. For oriented stars $F$ of degree exactly four, we show the only heroes in $F$-free digraphs are transitive tournaments, or possibly special joins of transitive tournaments. Aboulker et al. characterized the set of heroes of $\{H, K_{1} + \vec{P_{2}}\}$-free digraphs almost completely, and we show the same characterization for the class of $\{H, rK_{1} + \vec{P_{3}}\}$-free digraphs. Lastly, we show that if we forbid two "valid" orientations of brooms, then every transitive tournament is a hero for this class of digraphs.

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arXiv2023

The Kromatic Symmetric Function: A $K$-theoretic Analogue of $X_G$

Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with $K$-theory yields a rich combinatorial theory of inhomogeneous deformations, where Schur functions are replaced by their $K$-analogues, the basis of symmetric Grothendieck functions. We introduce and initiate a theory of the Kromatic symmetric function $\overline{X}_G$, a $K$-theoretic analogue of the chromatic symmetric function $X_G$ of a graph $G$. The Kromatic symmetric function is a generating series for graph colorings in which vertices may receive any nonempty set of distinct colors such that neighboring color sets are disjoint. Our main result lifts a theorem of Gasharov (1996) to this setting, showing that when $G$ is a claw-free incomparability graph, $\overline{X}_G$ is a positive sum of symmetric Grothendieck functions. This result suggests a topological interpretation of Gasharov's theorem. We then show that the Kromatic symmetric functions of path graphs are not positive in any of several $K$-analogues of the $e$-basis of symmetric functions, demonstrating that the Stanley-Stembridge conjecture (1993) does not have such a lift to $K$-theory and so is unlikely to be amenable to a topological perspective. We also define a vertex-weighted extension of $\overline{X}_G$ and show that it admits a deletion--contraction relation. Finally, we give a $K$-analogue for $\overline{X}_G$ of the classic monomial-basis expansion of $X_G$.

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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

Pure pairs. IV. Trees in bipartite graphs

In this paper we investigate the bipartite analogue of the strong Erdos-Hajnal property. We prove that for every forest $H$ and every $τ>0$ there exists $ε>0$, such that if $G$ has a bipartition $(A,B)$ and does not contain $H$ as an induced subgraph, and has at most $(1-τ)|A|\cdot|B|$ edges, then there is a stable set in $G$ that contains at least $ε|V_i|$ vertices of $V_i$, for $i=1,2$. No graphs $H$ except forests have this property.

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arXiv2023

The $r$-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $ν(G)$ denote the maximum size of a matching in $H$, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed $r, k, s \in \mathbb{N}$: $r$-Coloring in hypergraphs $G$ with edge size at most $k$ and $ν(G) \leq s$; $r$-Precoloring Extension in $k$-uniform hypergraphs $G$ with $ν(G) \leq s$; $r$-Precoloring Extension in hypergraphs $G$ with edge size at most $k$ and $ν(G) \leq s$; Maximum Stable Set in $k$-uniform hypergraphs $G$ with $ν(G) \leq s$; Maximum Weight Stable Set in $k$-uniform hypergraphs with $ν(G) \leq s$; as well as partial results for $r$-Coloring in $k$-uniform hypergraphs $ν(G) \leq s$. We then turn our attention to $2$-Coloring in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in $G$ implies that $ν(G)$ is bounded by a constant; and that $3$-coloring linear $3$-uniform hypergraphs $G$ with $ν(G) \leq 532$ is NP-hard.

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