Planar Digraphs of Digirth Five Are 2-Colorable
نویسندگان
چکیده
Neumann-Lara (1985) and Škrekovski conjectured that every planar digraph with digirth at least three is 2-colorable, meaning that the vertices can be 2-colored without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least five is 2-colorable. The result also holds in the setting of list colorings.
منابع مشابه
Planar digraphs of digirth four are 2-colourable
Neumann-Lara conjectured in 1985 that every planar digraph with digirth at least three is 2-colourable, meaning that the vertices can be 2-coloured without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least four is 2-colourable.
متن کاملA Construction of Uniquely n-Colorable Digraphs with Arbitrarily Large Digirth
A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets and we say a digraph is uniquely n-colorable when this decomposition is unique up to relabeling. It was shown probabilistically in...
متن کاملPlanar digraphs without large acyclic sets
Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note we show that for all integers n ≥ g ≥ 3, there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most dn(g−2)+1 g−1 e. When g = 3 this result disproves a conjecture of Harutyunyan and shows that a question of Albert...
متن کاملThe circular chromatic number of a digraph
We introduce the circular chromatic number χc of a digraph and establish various basic results. They show that the coloring theory for digraphs is similar to the coloring theory for undirected graphs when independent sets of vertices are replaced by acyclic sets. Since the directed k-cycle has circular chromatic number k/(k − 1), for k ≥ 2, values of χc between 1 and 2 are possible. We show tha...
متن کاملPlanar Graphs of Girth at least Five are Square (∆ + 2)-Choosable
We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five are square (∆ + 2)-colorable for large enough ∆. In fact, we prove the stronger statement that such graphs are square (∆+2)-choosable and even square (∆+2)-paintable.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Journal of Graph Theory
دوره 84 شماره
صفحات -
تاریخ انتشار 2017