11 research outputs found
D-colorable digraphs with large girth
In 1959 Paul Erdos (Graph theory and probability, Canad. J. Math. 11 (1959), 34-38) famously proved, nonconstructively, that there exist graphs that have both arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants (D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll and B. Mohar, The circular chromatic number of a digraph, J. Graph Theory 46 (2004), 227-240; B. Bollobas and N. Sauer, Uniquely colourable graphs with large girth, Canad. J. Math. 28 (1976), 1340-1344; J. Nesetril and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B 90 (2004), 161-172; X. Zhu, Uniquely H-colorable graphs with large girth, J. Graph Theory 23 (1996), 33-41) that have extended and generalized the result while strengthening the techniques used to achieve it. We follow the lead of Xuding Zhu (op. cit.) who proved that, for a suitable graph H, there exist graphs of arbitrarily large girth that are uniquely H-colorable. We establish an analogue of Zhu\u27s results in a digraph setting.
Let C and D be digraphs. A mapping f:V(D)&rarr V(C) is a C-coloring 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-colorable if it admits a C-coloring and that D is uniquely C-colorable if it is surjectively C-colorable and any two C-colorings of D differ by an automorphism of C. We prove that if D is a digraph that is not C-colorable, then there exist graphs of arbitrarily large girth that are D-colorable but not C-colorable. Moreover, for every digraph D that is uniquely D-colorable, there exists a uniquely D-colorable digraph of arbitrarily large girth
On Colouring Oriented Graphs of Large Girth
We prove that for every oriented graph and every choice of positive integers and , there exists an oriented graph along with a surjective homomorphism such that: (i) girth; (ii) for every oriented graph with at most vertices, there exists a homomorphism from to if and only if there exists a homomorphism from to C; and (iii) for every -pointed oriented graph with at most vertices and for every homomorphism there exists a unique homomorphism such that . Determining the chromatic number of an oriented graph is equivalent to finding the smallest integer such that admits a homomorphism to an order- tournament, so our main theorem yields results on the girth and chromatic number of oriented graphs. While our main proof is probabilistic (hence nonconstructive), for any given and , we include a construction of an oriented graph with girth and chromatic number
Uniquely circular colourable and uniquely fractional colourable graphs of large girth
Given any rational numbers and an integer , we
prove that there is a graph of girth at least , which is
uniquely -colourable and uniquely -fractional colourable
Uniquely D-colourable digraphs with large girth
Let C and D be digraphs. A mapping is a C-colouring if for
every arc of D, either is an arc of C or , 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 C-colourable
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 are D-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 , there are
uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of
Mathematic
Dualities and Dual Pairs in Heyting Algebras
We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categorie
Topos-like Properties in Two Categories of Graphs and Graph-like Features in an Abstract Category
In the study of the Category of Graphs, the usual notion of a graph is that of a simple graph with at most one loop on any vertex, and the usual notion of a graph homomorphism is a mapping of graphs that sends vertices to vertices, edges to edges, and preserves incidence of the mapped vertices and edges. A more general view is to create a category of graphs that allows graphs to have multiple edges between two vertices and multiple loops at a vertex, coupled with a more general graph homomorphism that allows edges to be mapped to vertices as long as that map still preserves incidence. This more general category of graphs is named the Category of Conceptual Graphs. We investigate topos and topos-like properties of two subcategories of the Category of Conceptual Graphs. The first subcategory is the Category of Simple Loopless Graphs with Strict Morphisms in which the graphs are simple and loopless and the incidence preserving morphisms are restricted to sending edges to edges, and the second subcategory is the Category of Simple Graphs with Strict Morphisms where at most one loop is allowed on a vertex. We also define graph objects that are their graph equivalents when viewed in any of the graph categories, and mimic their graph equivalents when they are in other categories. We conclude by investigating the possible reflective and corefective aspects of our two subcategories of graphs
On sparse graphs with given colorings and homomorphisms
AbstractWe prove that for every graph H and positive integers k and l there exists a graph G with girth at least l such that for all graphs H′ with at most k vertices there exists a homomorphism G→H′ if and only if there exists a homomorphism H→H′. This implies (for H=Kk) the classical result of Erdős and other generalizations (such as Sparse Incomparability Lemma). We refine the above statement to the 1-1 correspondence between the set of all homomorphisms G→H′ and the set of all homomorphisms H→H′. This in turn yields the existence of sparse uniquely H-colorable graphs and, perhaps surprisingly, provides a characterization of the graphs H for which the analog of Müller's theorem holds for H-colorings