Related papers: (Claw, C_3)-free digraphs with unbounded dichromat…
We show that digraphs with no transitive tournament on $3$ vertices and in which every induced directed cycle has length $3$ can have arbitrarily large dichromatic number. This answers to the negative a question of Carbonero, Hompe, Moore,…
The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded…
Motivated by an old conjecture of P. Erd\H{o}s and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable…
For $t \ge 2$, let us call a digraph $D$ \emph{t-chordal} if all induced directed cycles in $D$ have length equal to $t$. In a previous paper, we asked for which $t$ it is true that $t$-chordal graphs with bounded clique number have bounded…
The $3$-colorability problem is a well-known NP-complete problem and it remains NP-complete for $(claw, diamond, K_4)$-free graphs. Recently, $3$-colorability has been also considered for $(claw, N_{1,1,1})$-free graphs. Here, a generalised…
The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been…
We characterize all digraphs $H$ such that orientations of chordal graphs with no induced copy of $H$ have bounded dichromatic number.
An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph. The Gy\'arf\'as-Sumner conjecture is a widely-open conjecture…
A. Hajnal and P. Erd\H{o}s proved that a graph with uncountable chromatic number cannot avoid short cycles, it must contain for example $ C_4 $ (among other obligatory subgraphs). It was shown recently by D. T. Soukup that, in contrast of…
We prove that every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected. The result is sharp and it is inspired by a conjecture posed by Zheng, Broersma, Wang and Zhang in 2020.
We present an infinite family of 3-connected non-bipartite graphs with chromatic roots in the interval (1,2) thus resolving a conjecture of Jackson's in the negative. In addition, we briefly consider other graph classes that are conjectured…
This is the second in a series of two papers dealing with $(2P_3,C_4,C_6)$-free graphs, or equivalently, $(2P_3,\text{even hole})$-free graphs. In this two-paper series, we give a full structural description of $(2P_3,C_4,C_6)$-free graphs…
We introduce a new method to construct uncountably chromatic graphs from non special trees and ladder systems. Answering a question of P. Erd\H{o}s and A. Hajnal from 1985, we construct graphs of chromatic number $\omega_1$ without…
A {\it dynamic $k$-coloring} of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex of degree at least 2 in $G$ will be adjacent to vertices with at least 2 different colors. The smallest number $k$ for which…
We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least~5 in this class is…
In an attempt to understanding the complexity of the independent set problem, Chv{\'a}tal defined t-perfect graphs. While a full characterization of this class is still at large, progress has been achieved for claw-free graphs [Bruhn and…
We show that triangle-free graphs that do not contain an induced subgraph isomorphic to a subdivision of K4 are 3-colorable. This proves a conjecture of Trotignon and Vuskovic.
We show that there exist linear $3$-uniform hypergraphs with $n$ vertices and $\Omega(n^2)$ edges which contain no copy of the $3 \times 3$ grid. This makes significant progress on a conjecture of F\"{u}redi and Ruszink\'{o}. We also…
Let H be a tree. It was proved by Rodl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$ as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened…
This is the first in a series of two papers dealing with $(2P_3,C_4,C_6)$-free graphs, or equivalently, $(2P_3,\text{even hole})$-free graphs. In this two-paper series, we give a full structural description of $(2P_3,C_4,C_6)$-free graphs…