8 research outputs found
Colourings of -coloured mixed graphs
A mixed graph is, informally, an object obtained from a simple undirected
graph by choosing an orientation for a subset of its edges. A mixed graph is
-coloured if each edge is assigned one of colours, and each
arc is assigned one of colours. Oriented graphs are -coloured mixed graphs, and 2-edge-coloured graphs are -coloured
mixed graphs. We show that results of Sopena for vertex colourings of oriented
graphs, and of Kostochka, Sopena and Zhu for vertex colourings oriented graphs
and 2-edge-coloured graphs, are special cases of results about vertex
colourings of -coloured mixed graphs. Both of these can be regarded as
a version of Brooks' Theorem.Comment: 7 pages, no figure
Oriented Colouring Graphs of Bounded Degree and Degeneracy
This paper considers upper bounds on the oriented chromatic number, ,
of graphs in terms of their maximum degree and/or their degeneracy
. In particular we show that asymptotically,
where and . This improves a result of MacGillivray, Raspaud, and
Swartz of the form . The rest of the paper is
devoted to improving prior bounds for in terms of and by
refining the asymptotic arguments involved.Comment: 8 pages, 3 figure
On the existence and non-existence of improper homomorphisms of oriented and -edge-coloured graphs to reflexive targets
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
homomorphisms. We fully classify those oriented graphs with tree-width that
do not admit such homomorphisms and show that it is NP-complete to decide if a
graph admits an orientation that does not admit such homomorphisms. We prove
analogous results for -edge-coloured graphs. We apply our results on
oriented graphs to provide a new tool in the study of chromatic number of
orientations of planar graphs -- a long-standing open problem
On -chromatic numbers of graphs having bounded sparsity parameters
An -graph is characterised by having types of arcs and types
of edges. A homomorphism of an -graph to an -graph , is a
vertex mapping that preserves adjacency, direction, and type. The
-chromatic number of , denoted by , is the minimum
value of such that there exists a homomorphism of to . The
theory of homomorphisms of -graphs have connections with graph theoretic
concepts like harmonious coloring, nowhere-zero flows; with other mathematical
topics like binary predicate logic, Coxeter groups; and has application to the
Query Evaluation Problem (QEP) in graph database.
In this article, we show that the arboricity of is bounded by a function
of but not the other way around. Additionally, we show that the
acyclic chromatic number of is bounded by a function of , a
result already known in the reverse direction. Furthermore, we prove that the
-chromatic number for the family of graphs with a maximum average degree
less than , including the subfamily of planar graphs
with girth at least , equals . This improves upon previous
findings, which proved the -chromatic number for planar graphs with
girth at least is .
It is established that the -chromatic number for the family
of partial -trees is both bounded below and above by
quadratic functions of , with the lower bound being tight when
. We prove and which improves both known lower bounds and
the former upper bound. Moreover, for the latter upper bound, to the best of
our knowledge we provide the first theoretical proof.Comment: 18 page
On Chromatic Number of Colored Mixed Graphs
An (m, n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m, n)-colored mixed graph G to an (m, n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f (u)f (v) is an arc (edge) of color c in H. The (m,n)-colored mixed chromatic number x((m,n))(G) of an (m, n)-colored mixed graph G is the order (number of vertices) of a smallest homomorphic image of G. This notion was introduced by Nektfil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147-155). They showed that x((m,n))(G) <= k(2m n)(k-1) where G is a acyclic k-colorable graph. We prove the tightness of this bound. We also show that the acyclic chromatic number of a graph is bounded by k(2) k(2+) inverted left perpedicular log((2m+n)) log((2m+n)) k inverted right perpendicular if its (m, n)-colored mixed chromatic number is at most k. Furthermore, using probabilistic method, we show that for connected graphs with maximum degree its (m, n)-colored mixed chromatic number is at most 2(Delta-1)(2m+n) (2m + n)(Delta-min(2m+m3)+2) In particular, the last result directly improves the upper bound of 2 Delta(2)2(Delta) oriented chromatic number of graphs with maximum degree Delta, obtained by Kostochka et al. J. Graph Theory 24, 331-340) to 2(Delta-1)(2)2(Delta). We also show that there exists a connected graph with maximum degree Delta and (m, n)-colored mixed chromatic number at least (2m + n)(Delta/2)