Browsing by Author "Spirkl, Sophie"

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Coloring Algorithms for Graphs and Hypergraphs with Forbidden Substructures
(University of Waterloo, 2022-04-18) Li, Yanjia; Spirkl, Sophie
This thesis mainly focus on complexity results of the generalized version of the $r$-Coloring Problem, the $r$-Pre-Coloring Extension Problem and the List $r$-Coloring Problem restricted to hypergraphs and ordered graphs with forbidden substructures. In the context of forbidding non-induced substructure in hypergraphs, we obtain complete complexity dichotomies of the $r$-Coloring Problem and the $r$-Pre-Coloring Extension Problem in hypergraphs with bounded edge size and bounded matching number, as well as the $r$-Pre-Coloring Extension Problem in hypergraphs with uniform edge size and bounded matching number. We also get partial complexity result of the $r$-Coloring Problem in hypergraphs with uniform edge size and bounded matching number. Additionally, we study the Maximum Stable Set Problem and the Maximum Weight Stable Set Problem in hypergraphs. We obtain complexity dichotomies of these problems in hypergraphs with uniform edge size and bounded matching number. We then give a polynomial-time algorithm of the 2-Coloring Problem restricted to the class of 3-uniform hypergraphs excluding a fixed one-edge induced subhypergraph. We also consider linear hypergraphs and show that 3-Coloring in linear 3-uniform hypergraphs with either bounded matching size or bounded induced matching size is NP-hard if the bound is a large enough constant. This thesis also contains a near-dichotomy of complexity results for ordered graphs. We prove that the List-3-Coloring Problem in ordered graphs with a forbidden induced ordered subgraph is polynomial-time solvable if the ordered subgraph contains only one edge, or it is isomorphic to some fixed ordered 3-vertex path plus isolated vertices. On the other hand, it is NP-hard if the ordered subgraph contains at least three edges, or contains a vertex of degree two and does not satisfy the polynomial-time case mentioned before, or contains two non-adjacent edges with a specific ordering. The complexity result when forbidding a few ordered subgraphs with exactly two edges is still unknown.
Cycles and coloring in graphs and digraphs
(University of Waterloo, 2022-08-22) Hompe, Patrick; Spirkl, Sophie
We show results in areas related to extremal problems in directed graphs. The first concerns a rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture, made by Aharoni. The Caccetta-H\"{a}ggkvist conjecture states that if $G$ is a simple digraph on $n$ vertices with minimum out-degree at least $k$, then there exists a directed cycle in $G$ of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this well-known conjecture, namely that if $G$ is a simple edge-colored graph (not necessarily properly colored) on $n$ vertices with $n$ color classes each of size at least $k$, then there exists a rainbow cycle in $G$ of length at most $\lceil n/k \rceil$. In this thesis, we first prove that if $G$ is an edge-colored graph on $n$ vertices with $n$ color classes each of size at least $\Omega(k \log{k})$, then $G$ has a rainbow cycle of length at most $\lceil n/k \rceil$. Then, we develop more techniques to prove the stronger result that if there are $n$ color classes, each of size at least $\Omega(k)$, then there is a rainbow cycle of length at most $\lceil n/k \rceil$. Finally, we improve upon existing bounds for the triangle case, showing that if there are $n$ color classes of size at least $0.3988n$, then there exists a rainbow triangle, and also if there are $1.1077n$ color classes of size at least $n/3$, then there is a rainbow triangle. Let $\chi(G)$ denote the \emph{chromatic number} of a graph $G$ and let $\omega(G)$ denote the \emph{clique number}. Similarly, let $\dichi(D)$ denote the \emph{dichromatic number} of a digraph $D$ and let $\omega(D)$ denote the clique number of the underlying undirected graph of $D$. In the second part of this thesis, we consider questions of $\chi$-boundedness and $\dichi$-boundedness. In the undirected setting, the question of $\chi$-boundedness concerns, for a class $\mathcal{C}$ of graphs, for what functions $f$ (if any) is it true that $\chi(G) \le f(\omega(G))$ for all graphs $G \in \mathcal{C}$. In a similar way, the notion of $\dichi$-boundedness refers to, given a class $\mathcal{C}$ of digraphs, for what functions $f$ (if any) is it true that $\dichi(D) \le f(\omega(D))$ for all digraphs $D \in \mathcal{C}$. It was a well-known conjecture, sometimes attributed to Esperet, that for all $k,r \in \mathbb{N}$ there exists $n$ such that in every graph with $G$ with $\chi(G) \ge n$ and $\omega(G) \le k$, there exists an induced subgraph $H$ of $G$ with $\chi(H) \ge r$ and $\omega(H) = 2$. We disprove this conjecture. Then, we examine the class of $k$-chordal digraphs, which are digraphs such that all induced directed cycles have length equal to $k$. We show that for $k \ge 3$, the class of $k$-chordal digraphs is not $\dichi$-bounded, generalizing a result of Aboulker, Bousquet, and de Verclos in [1] for $k=3$. Then we give a hardness result for determining whether a digraph is $k$-chordal, and finally we show a result in the positive direction, namely that the class of digraphs which are $k$-chordal and also do not contain an induced directed path on $k$ vertices is $\dichi$-bounded. We discuss the work of others stemming from and related to our results in both areas, and outline directions for further work.
Foreshadowing the Grid Theorem for Induced Subgraphs
(University of Waterloo, 2024-08-07) Hajebi, Sepehr; Spirkl, Sophie
We prove several dichotomy theorems toward a complete description of the unavoidable induced subgraphs of graphs with large treewidth. This is motivated by the Grid Theorem of Robertson and Seymour (1986) which achieves the same goal for minors (and subgraphs). Given a graph class C, we say that C is clean if the only induced subgraph obstructions to bounded treewidth in C are the basic ones: complete graphs, complete bipartite graphs, subdivided walls and the line graphs of the subdivided walls. The analog of the Grid Theorem for induced subgraphs (still out of reach) is then observed to be equivalent to a characterization of all hereditary classes that are clean. We characterize all clean classes that are defined by finitely many excluded induced subgraphs. Specifically, we identify a family of “non-basic” obstructions which, in this scenario, litmus-test the clean classes against the non-clean ones. The analogous characterization remains elusive in the case of infinitely many forbidden induced subgraphs. Among the infinite sets of graphs whose exclusion is known to result in a non-clean class, the following four appear to expose a distinctive gap in our understanding of cleanness: (1) Graphs which are the union of three cycles, all sharing a vertex and otherwise pairwise vertex-disjoint. (2) Graphs which are the union of two vertex-disjoint cycles. (3) Graphs consisting of two non-adjacent vertices with three pairwise internally disjoint paths between them, known as thetas. (4) Cycles with an even number of vertices (at least four), known as even holes. For i = 1, 2, 3, 4, let Ci be the class obtained by excluding the ith set above. We prove a full “grid-type theorem” for each of C1 and C2. Both results extend to an arbitrary number of excluded cycles (instead of “three” and “two”) of lower bounded lengths. In C3 and C4, we characterize the “local” structure of graphs with large treewidth. Explicitly, given a graph H, we prove the following: (a) Every (theta, K3)-free graph of large enough treewidth has an induced subgraph isomorphic to H, if and only if H is a K3-free chordal graph (that is, a forest). (b) Every (even hole K4)-free graph of large enough treewidth has an induced subgraph isomorphic to H, if and only if H is a K4-free chordal graph. We generalize both (a) and (b) to the “right” class of Kt-free graphs for all t. We also derive, from a very special case of (b), one of the two conjectures of Sintiari and Trotignon on even-hole-free graphs of large treewidth.
Induced subgraphs and tree decompositions VI. Graphs with 2-cutsets
(Elsevier, 2025-01) Abrishami, Tara; Chudnovsky, Maria; Hajebi, Sepehr; Spirkl, Sophie
This paper continues a series of papers investigating the following question: which hereditary graph classes have bounded treewidth? We call a graph t-clean if it does not contain as an induced subgraph the complete graph Kt, the complete bipartite graph Kt,t, subdivisions of a (t x t)-wall, and line graphs of subdivisions of a (t x t)-wall. It is known that graphs with bounded treewidth must be t-clean for some t; however, it is not true that every t-clean graph has bounded treewidth. In this paper, we show that three types of cutsets, namely clique cutsets, 2-cutsets, and 1-joins, interact well with treewidth and with each other, so graphs that are decomposable by these cutsets into basic classes of bounded treewidth have bounded treewidth. We apply this result to two hereditary graph classes, the class of (ISK4, well)-free graphs and the class of graphs with no cycle with a unique chord. These classes were previously studied and decomposition theorems were obtained for both classes. Our main results are that t-clean (ISK4, wheel)-free graphs have bounded treewidth and that t-clean graphs with no cycle with a unique chord have bounded treewidth.
Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole.
(Elsevier, 2025-02) Chudnovsky, Maria; Hajebi, Sepehr; Spirkl, Sophie
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.
On coloring digraphs with forbidden induced subgraphs
(University of Waterloo, 2023-04-25) Carbonero, Alvaro; Spirkl, Sophie
This thesis mainly focuses on the structural properties of digraphs with high dichromatic number. The dichromatic number of a digraph $D$, denoted by $\dichi(D)$, is designed to be the directed analog of the chromatic number of a graph $G$, denoted by $\chi(G)$. The field of $\chi$-boundedness studies the induced subgraphs that need to be present in a graph with high chromatic number. In this thesis, we study the equivalent of $\chi$-boundedness but with dichromatic number instead. In particular, we study the induced subgraphs of digraphs with high dichromatic number from two different perspectives which we describe below. First, we present results in the area of heroes. A digraph $H$ is a hero of a class of digraphs $\mathcal{C}$ if there exists a constant $c$ such that every $H$-free digraph $D\in \mathcal{C}$ has $\dichi(D)\leq c$. It is already known that when $\mathcal{C}$ is the family of $F$-free digraphs for some digraph $F$, the existence of heroes that are not transitive tournaments $TT_k$ implies that $F$ is the disjoint union of oriented stars. In this thesis, we narrow down the characterization of the digraphs $F$ which have heroes that are not transitive tournaments to the disjoint union of oriented stars of degree at most 4. Furthermore, we provide a big step towards the characterization of heroes in $\{rK_1+K_2 \}$-free digraphs, where $r\geq 1$. We achieve the latter by developing mathematical tools for proving that a hero in $F$-free digraphs is also a hero in $\{K_1+F\}$-free digraphs. Second, we present results in the area of $\dichi$-boundedness. In this area, we try to determine the classes of digraphs for which transitive tournaments are heroes. In particular, we ask whether, for a given class of digraphs $\mathcal{C}$, there exists a function $f$ such that, for every $k\geq 1$, $\dichi(D)\leq f(k)$ whenever $D\in \mathcal{C}$ and $D$ is $TT_k$-free. We provide a comprehensive literature review of the subject and outline the $\chi$-boundedness results that have an equivalent result in $\dichi$-boundedness. We conclude by generalizing a key lemma in the literature and using it to prove $\{\mathcal{B}, \mathcal{B'} \}$-free digraphs are $\dichi$-bounded, where $\mathcal{B}$ and $\mathcal{B'}$ are small brooms whose orientations are related and have an additional particular property.
Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path.
(Springer, 2023-09-15) Scott, Alex; Seymour, Paul; Spirkl, Sophie
A graph G is H -free if it has no induced subgraph isomorphic to H. We prove that a P5-free graph with clique number ω ≥ 3 has chromatic number at most ωlog2(ω). The best previous result was an exponential upper bound (5/27)3ω, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erd˝os-Hajnal conjecture holds for P5, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P5-free graphs, and our result is an attempt to approach that.
Pure Pairs. VIII. Excluding a Sparse Graph.
(Springer, 2024-08-05) Scott, Alex; Seymour, Paul; Spirkl, Sophie
A pure pair of size t in a graph G is a pair A, B of disjoint subsets of V(G), each of cardinality at least t, such that A is either complete or anticomplete to B. It is known that, for every forest H, every graph on n ≥ 2 vertices that does not contain H or its complement as an induced subgraph has a pure pair of size (n); furthermore, this only holds when H or its complement is a forest. In this paper, we look at pure pairs of size n1−c, where 0 < c < 1. Let H be a graph: does every graph on n ≥ 2 vertices that does not contain H or its complement as an induced subgraph have a pure pair of size (|G| 1−c)? The answer is related to the congestion of H, the maximum of 1 − (|J | − 1)/|E(J )| over all subgraphs J of H with an edge. (Congestion is nonnegative, and equals zero exactly when H is a forest.) Let d be the smaller of the congestions of H and H. We show that the answer to the question above is “yes” if d ≤ c/(9 + 15c), and “no” if d > c.