Related papers: Long induced paths in $K_{s, s}$-free graphs

Let us say a graph is $s\mathcal{O}$-free, where $s\ge 1$ is an integer, if there do not exist $s$ cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when $s=2$, is not…

Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete…

We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $\Omega(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that…

In this paper we show that every graph of pathwidth less than $k$ that has a path of order $n$ also has an induced path of order at least $\frac{1}{3} n^{1/k}$. This is an exponential improvement and a generalization of the polylogarithmic…

An induced path factor of a graph $G$ is a set of induced paths in $G$ with the property that every vertex of $G$ is in exactly one of the paths. The induced path number $\rho(G)$ of $G$ is the minimum number of paths in an induced path…

In any graph, the maximum size of an induced path is bounded by the maximum size of a path. However, in the general case, one cannot find a converse bound, even up to an arbitrary function, as evidenced by the case of cliques. Galvin, Rival…

In 2012, Ne\v{s}et\v{r}il and Ossona de Mendez proved that graphs of bounded degeneracy that have a path of order $n$ also have an induced path of order $\Omega(\log \log n)$. In this paper we give an almost matching upper bound by…

The altitude of a graph $G$, denoted $f(G)$, is the largest integer $k$ such that under each ordering of $E(G)$, there exists a path of length $k$ which traverses edges in increasing order. In 1971, Chv\'atal and Koml\'os asked for…

We show that for $d\ge d_0(\epsilon)$, with high probability, the random graph $G(n,d/n)$ contains an induced path of length $(3/2-\epsilon)\frac{n}{d}\log d$. This improves a result obtained independently by Luczak and Suen in the early…

An extension of an induced path $P$ in a graph $G$ is an induced path $P'$ such that deleting the endpoints of $P'$ results in $P$. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced…

Graphs of bounded degeneracy are known to contain induced paths of order $\Omega(\log \log n)$ when they contain a path of order $n$, as proved by Ne\v{s}et\v{r}il and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray…

Given a triangle-free graph $G$ with chromatic number $k$ and a proper vertex coloring $\phi$ of $G$, it is conjectured that $G$ contains an induced rainbow path on $k$ vertices under $\phi$. Scott and Seymour proved the existence of an…

We prove that any bounded degree regular graph with sufficiently strong spectral expansion contains an induced path of linear length. This is the first such result for expanders, strengthening an analogous result in the random setting by…

We prove that every connected triangle-free graph on $n$ vertices contains an induced tree on $\exp(c\sqrt{\log n})$ vertices, where $c$ is a positive constant. The best known upper bound is $(2+o(1))\sqrt n$. This partially answers…

Consider a graph $G$ with a long path $P$. When is it the case that $G$ also contains a long induced path? This question has been investigated in general as well as within a number of different graph classes since the 80s. We have recently…

We present a modification of the Depth first search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies…

Paths P1,...,Pk in a graph G=(V,E) are said to be mutually induced if for any 1 <= i < j <= k, Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test…

Let $G$ be an infinite graph whose vertex set is the set of positive integers, and let $G_n$ be the subgraph of $G$ induced by the vertices $\{1,2, \dots , n \}$. An increasing path of length $k$ in $G$, denoted $I_k$, is a sequence of…

For a real c \geq 1 and an integer n, let f(n,c) denote the maximum integer f so that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in…

We show that every complete $n$-vertex simple topological graph contains a topological subgraph on at least $(\log n)^{1/4 - o(1)}$ vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph.…