相关论文: Oriented paths in n-chromatic digraphs

A path P(k,l,r) is an oriented path consisting of k forward arcs, followed by l backward arcs, and then by r forward arcs. We prove the existence of any oriented path of length n-1 with three blocks having the middle block of length one in…

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented…

We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the…

Cohen et al. conjectured that for every oriented cycle $C$ there exist an integer $f(C)$ such that every strong $f(C)$-chromatic digraph contains a subdivision of $C$. El Joubbeh confirmed this conjecture for Hamiltonian digraphs. Indeed,…

An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show…

A path with three blocks $P(k,l,r)$ is an oriented path formed by $k$-forward arcs followed by $l$-backward arcs then $r$-forward arcs. We prove that any $(2k+1)$-chromatic digraph contains a path $P(1,k,1)$. However the existence of…

Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for two…

We study the existence of oriented paths with two blocks in oriented graphs under semidegree conditions. A block of an oriented path is a maximal directed subpath. Given positive integers $k$ and $\ell$ with $k/2\le \ell < k$, we establish…

We show that for any natural number $k \ge 1$, any oriented graph $D$ of minimum semidegree at least $(3k- 2)/4$ contains an antidirected path of length $k$. In fact, a slightly weaker condition on the semidegree sequence of $D$ suffices,…

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying…

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such…

Oriented chromatic number of an oriented graph $G$ is the minimum order of an oriented graph $H$ such that $G$ admits a homomorphism to $H$. The oriented chromatic number of an unoriented graph $G$ is the maximal chromatic number over all…

We show that for any integer $k \ge 4$, every oriented graph with minimum semidegree bigger than $\frac{1}{2}(k-1+\sqrt{k-3})$ contains an antidirected path of length $k$. Consequently, every oriented graph on $n$ vertices with more than…

In new progress on conjectures of Stein, and Addario-Berry, Havet, Linhares Sales, Reed and Thomass\'e, we prove that every oriented graph with all in- and out-degrees greater than 5k/8 contains an alternating path of length k. This…

The dichromatic number of a directed graph is at most 2, if we can 2-color the vertices such that each monochromatic part is acyclic. An oriented graph arises from a graph by orienting its edges in one of the two possible directions. We…

The chromatic profile of a digraph $H$, denoted by $\delta_{\chi}^{+}(H,k)$, is the infimum $d$ such that any $H$-free digraph $D$ on $n$ vertices with minimum out-degree $\delta^{+}(D) \ge dn$ must be $k$-colorable. We determine the exact…

Let $\vec{K}_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of $\vec{K}_{\mathbb{N}}$ in which every monochromatic path has…

The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y, and (ii) how to orient the edges so as to…

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…

In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph $G=(V,E)$, an oriented $r$-coloring for $G$ is a partition of the vertex set $V$ into $r$ independent sets, such…