Uniquely D-colourable Digraphs with Large Girth
نویسندگان
چکیده
Let C and D be digraphs. A mapping f : V (D) → V (C) is a Ccolouring if for every arc uv of D, either f(u)f(v) is an arc of C or f(u) = f(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely Ccolourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that areD-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number r ≥ 1, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.
منابع مشابه
Uniquely circular colourable and uniquely fractional colourable graphs of large girth
Given any rational numbers r ≥ r′ > 2 and an integer g, we prove that there is a graph G of girth at least g, which is uniquely circular r-colourable and uniquely fractional r′-colourable. Moreover, the graph G has maximum degree bounded by a number which depends on r and r′ but does not depend on g.
متن کاملUniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth
For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k 12(g+1) vertices whose maximal degree is at most 5k 13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G) log jGj 13 log k and maximum degree (G) 6k 13 can exist. We also study several related problems.
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 کاملNote on robust critical graphs with large odd girth
A graph G is (k+1)-critical if it is not k-colourable but G−e is k-colourable for any edge e ∈ E(G). In this paper we show that for any integers k ≥ 3 and l ≥ 5 there exists a constant c = c(k, l) > 0, such that for all ñ, there exists a (k + 1)-critical graph G on n vertices with n > ñ and odd girth at least l, which can be made (k − 1)-colourable only by the omission of at least cn2 edges.
متن کاملEvery circle graph of girth at least 5 is 3-colourable
It is known that every triangle-free (equivalently, of girth at least 4) circle graph is 5-colourable (Kostochka, 1988) and that there exist examples of these graphs which are not 4-colourable (Ageev, 1996). In this note we show that every circle graph of girth at least 5 is 2-degenerate and, consequently, not only 3-colourable but even 3-choosable.
متن کامل