For a simple digraph $G$ without directed triangles or digons, let $β(G)$ be the size of the smallest subset $X \subseteq E(G)$ such that $G\setminus X$ has no directed cycles, and let $γ(G)$ be the number of unordered pairs of nonadjacent vertices in $G$. In 2008, Chudnovsky, Seymour, and Sullivan showed that $β(G) \le γ(G)$, and conjectured that $β(G) \le γ(G)/2$. Recently, Dunkum, Hamburger, and Pór proved that $β(G) \le 0.88 γ(G)$. In this note, we prove that $β(G) \le 0.8616 γ(G)$.
S.B. Rao conjectured in 1971 that graphic degree sequences are well quasi ordered by a relation defined in terms of the induced subgraph relation. In 2008, M. Chudnovsky and P. Seymour proved this long standing Rao's Conjecture by giving structure theorems for graphic degree sequences. In this paper, we prove and use a variant of Dickson's Lemma from commutative semigroup theory to give a short proof of the bounded degree case of Rao's Conjecture that is independent of the Chudnovsky-Seymour structure theory. In fact, we affirmatively answer two questions of N. Robertson, the first of which im…
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger's conjecture and the Erdős-Lovász Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all $k\ge7$, every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable; every even-hole-free graph $G$ with $ω(G)<χ(G)=s+t-1$ satisfie…
Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present paper we reprove their theorem using many of the same ideas, but with the key model-theoretic ingredient of first determining the so-called amount of stability of the graph. This approach changes the applicable Ramsey theorem, improves the bounds and offers a different structural perspective on the graphs in question. Complementing this, we give an infinitary proof which implies the finite result.
The second author's $ω$, $Δ$, $χ$ conjecture proposes that every graph satisties $χ\leq \lceil \frac 12 (Δ+1+ω)\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colourable complement. To prove our results we introduce a very useful $χ$-preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so-called "skeletal" graphs.
Chudnovsky, Scott, Seymour and Spirkl recently proved a conjecture by Kalai and Meshulam stating that the reduced Euler characteristic of the independence complex of a graph without induced cycles of length divisible by three is in {-1,0,1}. Gauthier had earlier proved that assuming no cycles of those lengths, induced or not. Kalai and Meshulam also stated a stronger topological conjecture, that the total betti numbers are in {0,1}. Towards that we prove an even stronger statement in the same setting as Gauthier: The independence complexes are either contractible or homotopy equivalent to sphe…
Chudnovsky and Seymour's structure theorem for claw-free graphs has led to a multitude of recent results that exploit two structural operations: {\em compositions of strips} and {\em thickenings}. In this paper we consider the latter, proving that every claw-free graph has a unique optimal {\em antithickening}, where our definition of {\em optimal} is chosen carefully to respect the structural foundation of the graph. Furthermore, we give an algorithm to find the optimal antithickening in $O(m^2)$ time. For the sake of both completeness and ease of proof, we prove stronger results in the more …
Let $G$ be a finite directed graph, $β(G)$ the minimum size of a subset $X$ of edges such that the graph $G' = (V,E \smallsetminus X)$ is directed acyclic and $γ(G)$ the number of pairs of nonadjacent vertices in the undirected graph obtained from $G$ by replacing each directed edge with an undirected edge. Chudnovsky, Seymour and Sullivan \cite{CSS07} proved that if $G$ is triangle-free, then $β(G) \leq γ(G)$. They conjectured a sharper bound (so called the "CSS conjecture") that $β(G) \leq \dfrac{γ(G)}{2}$. Nathanson and Sullivan verified this conjecture for the directed Cayley graph $\Cay(\…
We prove Engström's conjecture that the independence complex of graphs with no induced cycle of length divisible by $3$ is either contractible or homotopy equivalent to a sphere. Our result strengthens a result by Zhang and Wu, verifying a conjecture of Kalai and Meshulam which states that the total Betti number of the independence complex of such a graph is at most $1$. A weaker conjecture was proved earlier by Chudnovsky, Scott, Seymour, and Spirkl, who showed that in such a graph, the number of independent sets of even size minus the number of independent sets of odd size has values $0$, $1…
Given a positive integer $m\ge 3$, let $ch(m)$ be the smallest positive constant with the following property: \emph{ Every simple directed graph on $n\ge 3$ vertices all whose outdegrees are at least $ch(m)\cdot n$ contains a directed cycle of length at most $m$.} Caccetta and Häggkvist conjectured that $ch(m)=1/m$, which if true, would be the best possible. In this paper, we prove the following result: \emph{ For every integer $m\ge 3$, let $α(m)$ be the unique real root in $(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}. \end{equation*} Then $ch(m)\le α(m)$. This genera…
For a directed graph $G$ without loops or parallel edges, let $β(G)$ denote the size of the smallest feedback arc set, i.e., the smallest subset $X \subset E(G)$ such that $G \sm X$ has no directed cycles. Let $γ(G)$ be the number of unordered pairs of vertices of $G$ which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least $r \ge 4$ satisfies $β(G) \le cγ(G)/r^2$, where $c$ is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour, and Sullivan. This result can be also used to answer a quest…
The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time $O(n^4)$ whether three given vertices of a graph belong to an induced tree. Here, we study four-in-a-tree for triangle-free graphs. We give a structural answer to the following question: what does a triangle-free graph look like if no induced tree covers four given vertices? Our main result says that any such graph must have the "same structure", in a sense to be defined precisely, as a square or a cube. We provide an $O(nm)$-time algorithm that given a triangle-free graph $G$ together with four vertices outputs either an …
Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for every $P \in \mathcal{P}$. We say that $D$ satisfies the $α$-property if, for every maximum stable set $S$ of $D$, there exists an $S$-path partition of $D$, and we say that $D$ is $α$-diperfect if every induced subdigraph of $D$ satisfies the $α$-property. A digraph $C$ is an anti-directed odd cycle if (i) the underlying graph of $C$ is a cycle $x_1x_2 \cdo…
