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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 a recent center of study. In this work we look at possible extensions of Gyarfas-Sumner conjecture. More precisely, we propose as a conjecture a simple characterization of finite sets $mathcal F$ of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of $mathcal F$ as an induce subdigraph. Among notable results, we prove that oriented triangle-free graphs without a directed path of length $3$ are $2$-colorable. If condition of triangle-free is replaced with $K_4$-free, then we have an upper bound of $414$. We also show that an orientation of complete multipartite graph with no directed triangle is 2-colorable. To prove these results we introduce the notion of emph{nice sets} that might be of independent interest.
We define strongly chordal digraphs, which generalize strongly chordal graphs and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0,1 matrices that admit a simultaneous row and column permutation
We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Kellers conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. We present an aut
We prove a conjecture of Ohba which says that every graph $G$ on at most $2chi(G)+1$ vertices satisfies $chi_ell(G)=chi(G)$.
A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attem
Motivated by a hat guessing problem proposed by Iwasawa cite{Iwasawa10}, Butler and Graham cite{Butler11} made the following conjecture on the existence of certain way of marking the {em coordinate lines} in $[k]^n$: there exists a way to mark one po